The thing is though that even if Nixon were to call for a recount in Illinois and the state did go in his favor, that still isn't enough to overtake Kennedy in the electoral college. My guess is if a recount did somehow flip Illinois to Nixon and somehow flipped the national popular vote to Nixon in the process, Kennedy would go into his Presidency as illegitimate, just like GW Bush did after the Florida debacle of 2000.
I think that, in general, Nixon wouldn't have won because (regardless of the reality), the political machines in Illinois and Texas would have made it impossible to get at the truth. There WERE numerous cases of VERY clear voter fraud in both Illinois and Texas, including several cases where Kennedy received more votes in a precinct than there were voters, but the (all Democratic) state election board was quick to confirm the election even though they could see the disconnects.
If recounts HAD been conducted and resolved some of the issues, Nixon would probably have won Illinois, possibly the popular vote; probably not Texas. To win, the election, he would have had to win both Illinois and Texas. So, Kennedy would likely have won, but been somewhat weakened by a popular vote disconnect, clear signs that his supporters committed voter fraud on a large scale, and the support of mob bosses in Chicago.
So, Nixon may have been right not to call for recounts because:
a) The political machines would be very hard to overcome to get at "the truth"
b) Even if real recounts WERE done, it may not have given him victory
c) It would have made him look like a sore loser
In any case, either in 1960 or in 2000, a disconnect between electoral college result and popular vote does NOT make a President (or Presidency) "illegitimate." Many may say, or think it does, but that usually reflects a lack of understanding of our presidential election process. Other than that it drives the electoral votes, the nationwide popular vote is NOT a factor in selecting the president. That being said, if a president is elected, but loses the popular vote, it may create an IMPRESSION of illegitimacy or, at least, raise questions as to his "mandate."