This part of the theorem is true for sure. If p^2|M_q then q| p-1, which implies that

2^q - 1|2^(p-1) - 1, from which p^2|2^(p-1) - 1 follows. The reverse direction of that theorem (recall that the theorem is 'iff') is not at all obvious to me.

I'm not sure I follow you here. Can you explain? Remember that in my conjecture, the integer n in 2^n -1 can be composite, whereas in theorem quoted on wiki, n is a prime.