If JFK had signed the Civil Rights Bill, Nixon would have supported it. Nixon was not Goldwater.
What WOULD have happened was George Wallace and Strom Thurmond running as Dixiecrats, possibly even as the official Democratic nominees in at least some Southern states.
John F. Kennedy/Lyndon B. Johnson (D) 224 EV
Richard M. Nixon/James V. Rhodes (R) 211 EV
George C. Wallace/Strom Thurmond (Dixiecrat) 103 EVThis election would have gone to the House of Representatives, and THAT would have been interesting. It may have gone to the Senate.