JJ. Problematic hand vs known villain. NL50

[Jupit3r]

The villain is 29/19/3.5 over 450 hands. He's somewhat of a tricky player. He has AQs here most often but have seen him slow playing AA/KK pf. When he checks the turn I'm putting him on QQ+, 22, AQx. How's my line?

Cryptologic
No Limit Holdem Ring game
Blinds: $0.25/$0.50
5 players
Converter

Stack sizes:
UTG: $11.75
Hero: $49.80
Button: $117.30
SB: $94.56
BB: $122.06

Pre-flop: (5 players) Hero is CO with J J
UTG folds, Hero raises to $2, 2 folds, BB calls.

Flop: 2 J Q ($4.25, 2 players)
BB checks, Hero bets $4, BB raises to $8, Hero raises to $24, BB calls.

Turn: 9 ($52.25, 2 players)
BB checks, Hero is all-in $27.8, BB calls.

River: 8 ($107.85, 1 player + 1 all-in - Main pot: $107.85)


Results:
Final pot: $107.85
Hero shows Jc Jh
BB shows As Ad

[Fnord]

I would check the turn behind, however you pretty much committed yourself to this pot on the flop.

[Jupit3r]

I'm pretty sure I'm ahead on the flop, would a push be better play here?