He was “Jack” to his family and friends and this was widely publicly known. But at the time newspaper and broadcast reporting had quite a formal tone and his forename, when mentioned, would have been given as “John”. But I suspect in colloquial speech he was often called Jack Kennedy.
For some reason, there has stuck in my mind a couplet which was current in 1968, when Bobby Kennedy was seeking the Democratic nomination, which went something like:
“Jack was nimble, Jack was quick.
But Bobby makes me sick.”
Obviously, this was used by people who weren’t fans of Bobby Kennedy to compare him unfavourably with his (by then, sainted) brother. It suggests that both “Jack” and “Bobby” were current in colloquial speech to refer to the two men.