How did the term wigging out or he really wigged out in reference to someone going completely nuts? What connection is there even remotely to a wig?

Here's what the Online Etymology Dictionary has to offer:

wig 1670s, shortened form of periwig. Meaning "person who wears a wig (professionally)" is from 1828. The verb meaning "to behave hysterically" (usually with out) is attested from 1955, from notion in to flip one's wig. Cf. dash my wig!, a former mild imprecation (1797), also wigs on the green (1856), Irish colloquial for "a fight or rumble" (since wigs are likely to get detached from owners in such an event).

http://www.etymonline.com/index.php?search=wig&searchmode=none