Read Master of the Senate by Caro--fascinating account of ambition and canny, cunning political savvy.
I don't know if the story you heard was true--Robinson admired Nixon because Nixon, as President of the SENATE when he was Vice President under Eisenhower, was involved in the Civil Rights legislation efforts that were being waged in the Senate at that time--not much came of it--they got one piece of legislation (meaningless, but symbolic--voting rights, specifically) passed, but that one piece was like a crack in a dam, making the next efforts, years later, easier.
However, Robinson became disenchanted with Nixon quick, during the campaign, because Nixon was afraid to go to Harlem:
After baseball Robinson went to work for Chock Full O' Nuts as a spokesman then continued his efforts to advance civil rights. He became actively involved in the campaign for Republican candidate Richard Nixon in the 1960 Presidential elections. Robinson opted to support Nixon over John F. Kennedy because he liked the work that Nixon had done in the area of civil rights during Nixon's years as Vice President. However Robinson later described his regret on having supported Nixon.
Two incidents during the 1960 campaign were quite disillusioning to Robinson. In one incident Nixon was asked to comment on a statement by running mate Henry Cabot Lodge who stated that in a Nixon Administration a black would be named to the Cabinet; Nixon commented that Lodge was speaking on his own behalf. Later during the campaign Nixon refused to speak out when civil rights leader Dr. Martin Luther King, Jr., was locked in a full-security prison for a minor motor vehicle infraction.
Further Nixon refused to campaign in Harlem (while Kennedy did). These incidents drew Robinson a great deal of criticism from the African American community for his support of the Nixon campaign. By the end of the campaign the Kennedy ticket was looking more attractive to Robinson, but he had already committed to Nixon.
Jackie Robinson did establish good ties with New York Governor Nelson Rockefeller (who later became Gerald Ford's Vice President). Robinson supported Rockefeller's bid during the 1964 Republican Primiaries. However after the GOP ticket went to Barry Goldwater, Robinson was disgusted at what he saw during the 1964 Republican National Convention.
JFK had to be careful--he had the same shit on him that many Democrats have today--the "too" label. Too Eastern, too elitist, too rich, too liberal, too catholic...he was NOT universally beloved while he lived. In fact, he was a bit polarizing at times, like Clinton, even though he was also popular at times, especially when he was able to really pull out the savoir faire and make us look good as a nation. Jackie didn't hurt him, either--everyone liked looking at her.