I'm going to sound like one of those parents! long post
I realize I know my dog and his personality which may not be easy for someone else to experience upon first glance. I like to think of him as big and beautiful but others may not see him that way. However, anyone who's ever tried to pet him or know him quickly finds out he's a little bunny rabbit in a big dog suit.
I took him to a new vet for his annual checkup. He can't get vaccines due to past issues, but I still get the heartworm check and the routine physical/blood work.
The two attendants at the front desk both came over to meet him and welcome him. They pet him while he sniffed around and I filled out paperwork.
Then another attendant came and weighed him and took us to an exam room. She also talked to him reassuringly and then said she was taking him to the back to draw blood.
After a few minutes, she brought him back in and had a muzzle on him. I thought that was odd as I don't think he's ever been muzzled but I figured they may do it when they are doing the lab work. However, when the vet came in, they left it on. This made my dog very nervous and the attendant held him stiffly and I said I didn't think he liked being muzzled. The vet went up behind him and started listening to his heart and said "better safe than sorry". Then my dog growled a very low growl when she touched his back legs (which I admit, he does hate.) The vet said, "See? He's stressed." I said he stressed because he doesn't like the muzzle.
Now here's where I am going to sound like one of those parents who thinks their child does no wrong, lol. My dog has growled at a vet exactly ONCE in my presence prior to this and that was during a checkup after he had been hospitalized twice, had his stomach all shaved, and had three weeks of needles and tubes. I was shocked that time because just prior to this, two little boys had run up to him to pet him and the parents (and I) had freaked out a little cause we didn't know how he'd react when he wasn't feeling well. Meanwhile, my dog just laid there and let the little boys pet him and touch his nose.
But all checkups to this point have always been on a table and he hasn't been muzzled and the vets do their checking and looking in his ears and he doesn't like it but he has never done anything except whine a little.
Yesterday, when the vet was done with his back legs, she moved to the other side of the room and the attendant left the room. Then the vet said I could remove the muzzle and once I did, my dog started panting and doing what looks like the smiling face and he moved around and started sniffing again. He was relaxed. Then the vet asked ME to lift his gums and show her his teeth. I guess I was so surprised by all of this that I just did it but I was really thinking WTF? Are you afraid of THIS dog? This dog that will lick you to death if you let him? She never touched him or said anything to him. She stayed on the other side so I don't even know how well she could really see his teeth.
When we left to pay, a puppy pulled on his mom's leash and came right up to my dog and sniffed and wagged and my dog sniffed and wagged.
I will not take my dog to a vet that is scared of my dog! Am I overreacting?
= new reply since forum marked as read
I do remember one vet who seemed to be afraid of our rottweiler when we had one, but he wasn't our usual vet and no others ever had a problem with her.
This time, my dog did try to get near her but she kept her distance. I honestly do not think he was afraid of her. 
Loves 'em all. In the great Northern Wilderness of his ancestors people ate all the bad dogs.
Our little wild dog strives to be invisible even if humans are doing unspeakable things to her like sticking a thermometer up her butt, drawing blood, or giving her shots.
"Nope, I'm not here, this isn't happening! No eye contact! No eye contact! Look away!"
Our oldest most domestic mild-mannered sweetest dog in every way, that's the one who is most trouble at the vet's. She recently had a tumor cut out of her, benign thankfully, but if she'd come back with a muzzle, or the vet asked me to muzzle her, I'd understand. Certain things terrify her into mindlessness, and going to the vet is one of those things.
The sound of diesel engines is another. As a puppy she was picked up by animal control at a truck stop. I'll never know what horrors she experienced.
She has some issues, not just diesel engines. We were her last resort "rescue parents" and for a couple of years she didn't really know how to be a dog. Our other dogs, her mentors, would sometimes look at her like
"Just be a dog!"
She's grown up to become a pretty good dog, she's mellow now and a bit of a clown, but it took some time.
his sister, our black lab that we lost last year, was the diva and ruled the roost and clearly told this one what to do all the time. He is still lost without her.
He will try to climb into the chair with you if there's a thunderstorm and he is bigger than the chair. 
Just because your dog was fine with the old vet and the people there that he knew doesn't mean he'll be ok with the new place or new people. The dog did something that made them decide he should be muzzled... growled, showed his teeth, something. They're right in putting on the muzzle to keep themselves safe... they don't know your dog and even if they did they should be safety conscious of themselves if the dog shows any sign that he might snap. Being comfortable with kids or other people in normal surroundings doesn't mean a dog is going to feel comfortable with new people at the vet's office. He can smell the discomfort and distress of the other animals there, and this doesn't make any dog feel comfortable just walking in the door. He may have gotten accustomed to your old vet because over time he got to know the place, the people, and decided it wasn't a scary place for him.
Closed mouth muzzles suck, and they just make a dog feel even more anxious not allowing them to pant and especially if they never had to wear a muzzle before. Don't blame the vet for being safety conscious. Your dog did something to warrant they're not feeling safe without the muzzle. It's no big deal and hardly a reason to not want to use the vet anymore. I don't know what happened to your old vet that caused you to go to this new one, but you can't expect your dog to feel comfortable in this new place with people he doesn't know poking and prodding him.
In time your dog may feel more comfortable going to this place and grow accustomed to these people. However, every dog should have an open mouth muzzle just in case that you can bring with you for vet appointments should it be felt it's needed. You never know what might trigger aggression because of whatever the vet or techs need to do especially if they have to poke and prod painful areas... dogs are normally protective of their injuries or where they have pain for some other reason. It's a rare dog that doesn't have a problem going to a strange vet being prodded in places where they have a hurt.
I use a Jasco plastic muzzle for my dog and bring it with me for every vet visit just in case. Though he's not a problem at the vet's office he doesn't like his back end being probed though often he can be sufficiently distracted for needle pokes so much so that he doesn't even notice them, but normally he hates needles. The Jasco muzzle is a hard plastic that lets the dog freely open his mouth to pant or even drink water once they get used to wearing it. Since I knew there was probably going to be a time where Yoshi might need to wear the muzzle at the vet I measured him and it turned out that Boo's old muzzle was the right fit for him, too, and got him accustomed to having it put on and wearing it at home long before he ever needed to wear it at the vet. I bought it at the Leerburg site, and I think that's how I first found Leerburg because I had one hell of a time finding a Jasco muzzle anywhere. They have a new muzzle now that I think I might like better, but haven't gotten it since the Jasco one works well, and he rarely needs to have it on anyway. Leerburg also has loads of instruction that shows you exactly how to measure your dog to make sure you get the right size because the correct size is important for any muzzle.
Check out the different open mouth ("cage" or "basket"
muzzles and seriously consider getting one for your dog and practicing at home to get him accustomed to wearing it so it doesn't wig him out. Boo went absolutely nutso when they put a closed mouth muzzle on him once to take an x-ray, and being a hairy beast he needs to be able to have his mouth open to pant all the time. Yoshi is the same hairy breed that almost never has his mouth closed, so there's no way I'd want him to go through the trauma that Boo did just the few minutes he had to wear a closed mouth muzzle. I can't even understand why they even make closed mouth muzzles.... what dog wouldn't panic wearing that?
Leerburg also has good instructions for getting your dog used to having the muzzle put on, taken off and wearing it. Though Yoshi would certainly prefer not to wear the muzzle, he's fine with having it put on and wearing it. What I did was first just let him sniff it, and I'd use it to pet him so he got comfortable with this new thing just touching him. That was it for the first day. I did that again the second day, but after that I put a tiny dab of peanut butter inside the muzzle right were his nose would be, and he stuffed his face right in the muzzle himself to lick it off. I didn't strap it on him that time but just held it in place with my hand, and when he was done with the peanut butter licking just took it off, and that was it for that day. The next day I went through the same ritual but this time I strapped the muzzle on while he was enjoying the peanut butter and just sat back. It took him a little while to figure out that it was stuck on his head. I just left him like that for a minute and pet him and giggled (he's happy when I make giggling or laughing sounds) and then took it off and praised him. Same thing the next day but left the muzzle on a little longer. When he tried to paw at it, I told him "no" and distracted him with tickles and hugs and pets for a few minutes then took the muzzle off and praised him really well and gave him a treat. He's been fine with the muzzle ever since and will stuff his face in it just in case there's peanut butter in there since when we practiced it was.... once a dog thinks there might be something yummy somewhere they'll continue to think there might be even if it never is again in that same spot. Yoshi ALWAYS stuffs his own face in his muzzle on the off chance that there MIGHT be peanut butter in there just because a couple of years ago a couple of times there was. I have no doubt he'll keep doing that forever just in case there is though once in a blue moon I may put a little dab of peanut butter in there again just to confirm for him that he should always check.
Check out the various muzzles here (the cloth muzzles are closed mouth ones, so don't bother with them). I didn't like the wire basket muzzles because they weren't big enough to really let him open his mouth to pant, and I didn't trust he wouldn't bend those wimpy little wires. I also like the Jasco plastic muzzle since it's a snap to clean. You can get it in black or white or clear, but I like black best since the white and clear ones gets stained, scuffed and dirty looking no matter how much you clean it...
http://leerburg.com/muzzle.htm
Here's where it tells you how to measure your dog's snoot for the right size muzzle with video, text, photos and size charts...
http://leerburg.com/muzzlefit.htm
One thing I really like about Leerburg is not only to they test all their products themselves but they buy only US made and preferably union made products. They also use edible packing peanuts to ship their products that dissolve in water quickly so they won't cause a bowel obstruction in case your pet or your kid eats one and won't cause any landfill environmental problems like the styrofoam peanuts that most companies use...
http://leerburg.com/flix/player.php/71/Leerburg_Shipping_Peanuts_-_YOU_CAN_EAT_THEM/