Or very little, anyway. They're used to the way things are, and the current situation is working okay for them, so they don't see a problem. Until one day, our jacked up health care "system" bites them or someone they personally care about on the ass.
Maybe they lose the job that they've always relied on for health insurance, and are surprised at how hard it is to find another one with equally good benefits, or they need a treatment their insurance company doesn't want to approve. Or the copay for a drug they can't survive without suddenly goes through the roof. The list goes on and on, but the bottom line is, something they've been taking for granted suddenly just isn't there, and they realize what they've been relying on was a flimsy house of cards all along instead of the rock solid guarantee they thought it was. And they go, "Wow, this really is messed up, isn't it?"
Add to that the fact that way too many Americans are ill-informed about how health care works in other countries compared to the U.S., or how our health care falls short. Too many people believe that America has the "best health care system in the world," because that's what they've been told all their lives, and get all huffy if you try to explain why it actually isn’t. What's really needed, imo, is a massive public education program. But that costs money, so good luck with that. Especially since the insurance companies have loads of money to spend on lobbying to keep things the way they are. 
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