Party Poker $400 NL ten-handed

Stacks
Hero $600
Villain $600

I'm in late position with 44
Villain seems to be a loose player.

Villain raises $26 from middle position,
I call, blinds fold.
FLOP comes Q42 giving me middle set.
Hero checks, Villain bets $56 (size of pot).
Hero calls, thinking villain has top pair, probably AQ.
TURN is Q, Villain bets $100, Hero raises $100 more. Villain calls.

At this point, I have a full house, and I figure the villain is going to lose his stack with his trip queens.

Now on the river, a bad card falls...

RIVER: A
Villian bets all in for $200 remaining.

Should I lay down the hand? My read says this guy just made a better full house (with QQQAA vs my 444QQ)

I called, and the villain showed AQ, having filled up on the river. Was this a donkey call? Or a must-call situation? ($200 to call a $1000 pot)

Did I screw up?

PLease tell me your thoughts on this hand. Should I have gone all-in on the TURN? I wanted to get as much out of him as possible, and with such a strong hand, no reason to scare him away... bad play?