Jack is a diminutive form of the name John. During the Middle Ages you will find many nicknames bearing only slight resemblance to their formal counterparts, some survived until today and some became rather obsolete. For example Peggy (or Daisy)-Margaret, Bill-William, Hank/Harry-Henry, Sally-Sarah, etc. There are various explanations as to how these nicknames were derived. Many third or fourth-hand sources, the most reliable being the book Family Names from the Irish, Anglo-Saxon, Anglo-Norman and Scotch, say John was pronounced Jen (like modern French Jean). Adding the Norman pet form “kin” produced Jenkin—which morphed into Janken, then Jackin, then Jack.

According to many sources including John F. Kennedy's estate museum, (http://www.nps.gov/jofi/faqs.htm) his family called him "Jack" as a shorthand for "John"