Given the blinds, your position, and your stack you should either be pushing preflop or folding preflop. A min raise in this situation creates an altogether bad situation as you've seen.

For one thing, the BB is getting ~3:1 to call you if nothing else. Those are pretty tempting odds and I'd be inclined to move in with any ace against your weak raise.

Like mentioned before, your best case scenario on this call is being on the ass end of a coin flip. That's if you're lucky and he has a pocket pair under QQ. Otherwise, the guy isn't pushing with worse than KQ. Don't be seduced by that hand. It looks pretty at first but you'll lose a lot playing it into raises.

Just consider the possible hands he'd make this play with:

AA, KK, QQ - you're a total dog.
JJ through 66 - you're on the idiot end of the coin flip.
AK, AQ, AJ, AT, Ax - you're either dominated or you're behind.

None of those look very good to me. That's why I'm folding preflop.