checking the flop was a bad decision because check calling sucks we may as well just text the guy and tell him we have specifically JJ, QQ, KK, AJ or AQ. We have made an aggressive action preflop (squeeze doesn't sound horrific at all btw PF) and we've got a board that hits the pre-flop aggressors range harder than anyone else in the hand, for this reason we should bet.
The fact that we have top pair when it's a board everyone expects us to bet if we DIDN'T have top pair is another very good reason why we should bet.
If you get yourself into these spots post-flop my making lighter squeezes/3bets or even opens then you
have to be c-betting the boards that hit you hardest with all your range, decided to bet here with TT/AA/AK and then with

or

but not with AJ, A9 or even A3s should you have chosen to squeeze it is a massive error as your c-betting rrange in these spots is now very polarized, giving smart players every opportunity to bluff, value-bet or even hero-fold vs you quite easily and make pretty simple exploitative adjustments to their strategy in these pots to basically just completely own you.
AND +1, hilarious OP
