Folding JJ to A on the turn

Poker Stars
No Limit Holdem Ring game
Blinds: $0.10/$0.25
5 players
Converter

Stack sizes:
UTG: $24.75
CO: $48
Button: $29.70
Hero: $24.65
BB: $31.35

Pre-flop: (5 players) Hero is SB with J:heart: J:diamond:
2 folds, Button calls, Hero raises to $1.25, BB folds, Button calls.

Flop: 6:heart: 3:club: 7:club: ($2.75, 2 players)
Hero bets $2.1, Button calls.

Turn: A:heart: ($6.95, 2 players)
Hero checks, Button bets $3.75, Hero folds.
Uncalled bets: $3.75 returned to Button.

Results:
Final pot: $6.95

Villain is 44/7, pretty loose passive.

Bet out on the turn?

Yes, you must bet the turn. Check folding is pretty bad. If he raises it's an easy fold, but the board is very draw heavy and there is no reason to instantly give him credit for an ace. He may have peeled with Ax or something, but you need to bet to protect what is likely the best hand.