As the previous poster said, the spelling
Zane can be an Italian pet form of
Giovanni, which is of course a variation of
John.
Unfortunately, the use of
Zane as a given name in the USA is NOT from that Italian name but comes from the author
Zane Grey, who was descended from the American
Zane family, which arrived in colonial New Jersey from England. No one has been able to trace them back to
Europe and figure out what their surname originally meant, but it is probably Engliah and not Italian and so one cannot assume it's related to "
John."