Nixon never made an effort to repeal the 22nd amendment. He was disgraced and resigned from office. He had no chance of being elected again.

Ronald Reagan did, however and this was before the Star Wars prequels came out, so he might have just mixed up his former Republican presidents.

Source:

http://www.nytimes.com/1987/11/29/us/reagan-wants-end-of-two-term-limit.html

I have a question related to this as well. In my undergrad days my professor (who focused on the American Presidency and Nixon in particular) stated that the 22nd amendment was pushed by Republicans to prevent another FDR 4 term scenario. I remember said professor stating that there was an effort in the 1980's to repeal this amendment in the hopes that Ronald Reagan would run for a third term in 1988. I've never seen any evidence to corroborate this. Did such a movement exist?

No, he never did, and he wasn't planning on doing so. Nixon is one of the historical characters I've had an "obsession period" with, and I never have seen any evidence of that. It's not for a lack of grandiose thinking on his part, believe me: it either never crossed his mind or he was too realistic about his chances of doing so. Nixon was respected, not loved, and he was the first person on Earth who'd tell you that. There's also the fact that Pat Nixon didn't like politics all that much by 1972, and whatever else you can say about Tricky Dick, he really, really loved his wife. He likely would respect her wishes after 8 years of the Presidency. Even if he wanted to, I don't think he would have ever gotten the considerable amount of popularity needed to overturn the 22nd, or win the 1976 election if he did. Let's not forget we are talking about a full blown change to a Constitutional Amendment here.

The only post-war Presidents (Obama included) who I think even stood a chance of doing that, in my opinion, were Eisenhower, Reagan, and Clinton. All three wanted it to some extent, Reagan and Clinton because they absolutely loved the job and it showed, Eisenhower because he deep down didn't trust any of the possible successors fully (with the possible exception of Robert B. Anderson, Eisenhower probably didn't think any of the major political figures at the time could fully step into his shoes in a Cold War context. He might have been right in a certain sense. Rockefeller, Nixon, Stevenson, Kennedy, Johnson, he had different but undeniable problems with all of them). But even they wouldn't have a slam dunk of it: Eisenhower and Reagan were running into age/health problems and had Democrat dominated Congresses. However much Eisenhower and Reagan were respected by Democrat opponents (unlike Nixon), they wouldn't be able to get them to sink their own party's chances of winning. Kennedy and Dukakis would have definitely preferred Nixon and Bush to Eisenhower and Reagan. And they weren't omnipotent. Eisenhower had to deal with a general desire for change, economic troubles (mild by our standards, but still there), and a couple of (again, mild by our standards) scandals/policy failures. Reagan, while less affected than Bush Senior and still generally highly popular, did have the ghosts of Iran-Contra hobbling his final days and was nowhere near the demigod that he would become in GOP legend.

Clinton was relatively youthful and vigorous compared to Eisenhower and Reagan-that wasn't a problem. But he was in the same situation as Reagan politically in some ways: while still highly popular and (also like Reagan) presiding over a time of massive prosperity and good feelings, he had to deal with the fallout from the Lewinsky scandal. And unlike Eisenhower or Reagan, he was probably too divisive with the other party at the time to even attempt to grease it through Congress. In spite of the GOP overplaying their hand in the 1998 midterms, it was still a Republican majority Congress, and a Republican Party that loathed him.

And all three men had perfectly legitimate Vice Presidents who all publicly wanted the job. All three were perfectly qualified for it, and there was no reason to not let them run-one of them (George Bush Senior) was probably even more qualified than the President was.

Still I do think Eisenhower would have beaten Kennedy-who barely managed to beat the weakened Nixon (not exactly the most uniting or loving character in American politics) and that was with not so subtle help from just about every influential intellectual/media circle in the US and "a few honest crooks" in Illinois and Texas. Reagan would have beaten Dukakis, who lost in a landslide to Bush Senior, a man had problems within his own party, was hardly a charismatic political nut-cutter like Ronnie, and was hit far harder by Iran-Contra than Reagan. And Clinton would have beaten Bush Junior, who managed to just barely beat Gore-who like Nixon, was weakened, not least by his own boss, and also was in disadvantageous terrain-and required some opaque maneuvers in the highest levels of government, with a lot of "special action" in Florida to win. He even required the most obscene tactics imaginable from Karl Rove and Company to beat McCain in the primary. The problem is getting to that point. I think the opposition party would sink such an idea.

This is just speculation and it can be nothing but speculation, as a disclaimer. I honestly don't think JFK would have been able to overturn the 22nd either, had he lived. He wouldn't have been as unpopular as Johnson was in 1968, but probably not at Reagan/Clinton levels of popularity. And definitely not the demigod he's been turned into with the assassination.