Checking my reasoning

Had this hand pop up last night, posted it in my operations thread but thought I'd post it here too to get some feedback on if my reasoning was right in regards to the play.

BB is 56/30, MP2 is 12/2

Full Tilt No-Limit Hold'em, $0.50 BB (9 handed) Hand History Converter Tool from FlopTurnRiver.com (Format: FlopTurnRiver)

saw flop|saw showdown

CO ($83.80)
Button ($75.65)
Hero ($63.20)
BB ($45.45)
UTG ($58.45)
UTG+1 ($14.55)
MP1 ($39.50)
MP2 ($39.60)
MP3 ($50.50)

Preflop: Hero is SB with Q, Q.
4 folds, MP3 raises to $2, 2 folds, Hero calls $1.75, BB raises to $13, MP3 folds, Hero raises to $39, BB raises to $58.45 (All-In), Hero calls $8.45.

Flop: ($96.90) A, 4, 8 (2 players, 1 all-in)

Turn: ($96.90) 4 (2 players, 1 all-in)

River: ($96.90) A (2 players, 1 all-in)

Final Pot: $96.90

When MP2 raised I had him on probably AQ+ given his stats over 750 hands so I was willing to take a flop and see how it played out from there. When the BB then re-raised 6x my first thought was a squeeze play by him and I was willing to play back at him, depending on what MP2 did.

If he called, I'd have called to still go along with the set theory (though maybe that would be a place for another squeeze) and if he folded like he did then I was willing to play for stacks with the BB. My main thinking here was that the bet sizing just seemed wrong...if he had of had KK/AA then I can't see how he wouldn't have wanted to bet only around 3x the raise to try and keep a weaker hand in, if not limping along for trapping.

If he had of raised 3x then I probably would have been done with the hand, and yes I know people will look at his stats and say I should be willing to felt it there but wouldn't a normal size re-raise into a raiser and cold-caller indicate more strength? I'm not sure but in this case it just seemed wrong in bet sizing and right in circumstances for it to be anything but a squeeze.

Just threebet preflop and all this won't be necessary.

In your situation I prolly stack off given BB's retarded stats and big reraise.

Why 3-bet pre though given we are oop and can pretty well define MP2's range to be more than likely ahead of us?

If we 3-bet and he calls then calls down an A flop do we just shut down and let them take control?

I don't have a problem with my play while MP2 was in the hand, my question was if my reasoning for thinking BB was squeezing was correct, but could also add in what to do if MP2 reraises him, do we fold? (in my mind the answer would be yes)