Well, if there’s Nixon with Kennedy as Vice President, and Bill Clinton meets either one, does he also become a Republican? Or does he still become a Democrat? What about Hillary? Does she stay a “Goldwater girl” (or Nixon girl since he’d be president) or still come to support McGovern in ‘72? (that is if he still runs.) What about Bobby and Ted? Does Bobby still seek public office in New York or does he move out to a diffrent state? Could Bobby become president in the 80’s or 90’s? He’d only be in his 60’s and early 70’s...
A) Bill Clinton probably still becomes a Democrat no matter what happens to JFK. Arkansas was a Democratic state when he entered politics there in the 1970s, plus he simply was aligned with Democratic values. And Richard Nixon, who he would meet instead, was by no means the inspiring figure that Kennedy was...
B) HRC is difficult to determine. She would definitely support Nixon at first, but I'm of the opinion that Nixon would be a one termer. He would've invaded Cuba in 1961 after the Bay of Pigs (he told JFK to do this in OTL and got angry when Kennedy disagreed), and I don't think that conflict would've ended well especially when one considers that Nixon is likely to strike Laos and even Vietnam during the same period. (Laos in fact was considered more important than Vietnam in 1961, and Eisenhower told Kennedy to invade. Given Nixon's war policies in OTL, he probably would've listened). With at least two wars that are not unlikely to descend into guerrilla warfare by 1962, I think Nixon would be unpopular come 1964. In OTL HRC supported McCarthy in 1968 even as she was still a Republican. I think she supports the Democrats in 1964 and may still end up becoming a Democrat as a result.
C) If Jack's career sputters out in the 60's, I highly doubt that any of the Kennedy Bros would get elected President.
That said, people so far have given good reasons why the Kennedy's didn't become Republicans. I would add that even if Joe Sr had become a Republican in the 1920s, he would still support FDR in 1932 because he thought Roosevelt was the one man who could save capitalism. He wouldn't have been alone - Henry Wallace was a prominent Republican businessman who crossed party lines to support Roosevelt in '32.