In 1960, there were 14 unpledged electors, and one faithless elector, from the South who, in OTL, voted for Harry F. Byrd as a protest vote against desegregation. In OTL, it did not matter who they voted for, as Kennedy had enough pledged electors to win the electoral college. However, all it takes is a 0.6% shift in the popular vote...
Actually only 0.3% shift from Kennedy to Nixon. Remember that 1 flipped vote makes a change of 2 votes in the difference. If the vote is A 53 B 47, A wins by 6. Change 1 vote: A 52 B 48, and A only wins by 4.
... nationwide, to give Richard Nixon Hawaii, Illinois, and Missouri. This shift makes the electoral vote 263 for Nixon vs. 260 for Kennedy, with 14 electors unpledged to either candidate. If it comes down to this, which candidate will be able to convince the faithless electors ...
ITYM "unpledged", not "faithless". And that 15th Byrd elector isn't faithless until he has voted.
What do you think would happen?
If there is an 0.6% shift in the popular vote, Nixon wins the PV by 600,000 votes, with a narrow
majority, and leads in the EV as well. That will create pressure to elect Nixon. I don't think either ticket would want to deal with the Dixiecrat electors.
If no deal is made, the election goes to the House. The Democrats control 29 delegations, and even a 0.6% swing isn't going to flip many seats - much less enough to flip enough delegations to give the Republicans a majority.
The Democrats could just say that the election was close, the Constitution gives the choice to the House, and then just vote in Kennedy.
However, the Democrat delegations of course include all 11 from the South... so the bargaining could continue. If the Republicans hold out, and several Southern delegations abstain (or vote for Byrd), then Johnson, elected by the Senate, would take office on 20 January. (Note that the Senate gets only two choices, not three.)
Another "however" is that Nixon got a majority of the PV, and it will be argued that the House should honor the majority's wishes. That might be why Republicans refuse to vote in Kennedy.
An outside possibility - Nixon (probably not Kennedy) goes to some of the unpledged electors (he only needs the six from Alabama), and deals for their votes with something
not involving civil rights - cotton subsidies, or some big piece of pork. Sleazy, but just ordinary politics, not selling out an important moral issue. I think people would be relieved.
A similar deal in the House would be much harder, as the Republicans control only 17 delegations; they would have to buy 9 Southern delegations somehow.