(my first post so please be gentle)

Party 10+1 SNG 3 places paid 4 players remaining

Hero ~1200t
Button ~2300t
SB ~1300t
BB ~3200t

Button has been loose and active both before and after flop; SB is weak-tight; BB is solid. Blinds are 50/100.

Hero picks up QQ and bets 200.
Button pushed all in.
SB folds.
BB pushed.

Hero pushed. But afterward, I got to wondering if going all in was good tactics but bad strategy. I'm not sure of the math, but it looks like if they both have underpairs (best case), there's about a 30% chance that one of them will make a set when I do not. If one has AK and the other an underpair, my chances are worse.

What do you think - should I fold and try to make the money or take my chances with what is probably the best hand?