According to Wiki (Wieferich prime - Wikipedia, the free encyclopedia), for a Mersenne number M_q, if there exists a p^2|M_q where p and q are relatively prime, then p is a Wieferich prime. This result does not seem trivial to me, and it is uncredited, so I wonder if it is correct. It is equivalent to saying that not only are Wieferich primes counterexamples to your conjecture, but they are the only counterexamples. If this is correct, then you can prove there are only finite Wieferich numbers by placing a lower bound on your conjecture.