Villain is probably a regular, I had no reads on him and not enough hands to deduce anything. The effective stacks were 116 BB. The pot odds were 1.58:1 to call, that means I need 39 % equity to call. Assuming his range is AKs and JJ+ (which is a stretch based on my perception) I have 42 % equity with QQ. So I am pretty confident that the fold is standard play and not nitty based on the population tendencies. What do you think about my 4bet sizing? What sizing do you recomend?

There is very few regulars at 2NL that 5bet jam anything else than KK+. It sucks, but it's a fold (at this specific limit).

4b sizing's fine.
I think folding's fine though I'm sure others will disagree.
I'd rather flat the 3b than 4b/fold.

+1 to flatting the 3 bet from these positions.

dont 4b if youre folding especially ip.....

^ Yeah, don't 4-bet QQ if Villain shoves or folds and only shoves with AK+, it is now 72o.

call the 3bet my man

Vs a squeeze by BB, I just call. Readless, I don't even 4-bet kings in that spot. (Not because I'm particularly worried about aces when villain 3-bets, but because it's so hard for villain to 5-bet with worse. I think you make more money flatting and letting him c-bet his air.)

Calling the 3bet makes perfect sense. Should I do it with all of my premium holdings then? What if I was OOP?

When you're OOP it's more important to re-raise. (e.g. I almost always 4-bet QQ in UTGvBTN, although I might just call if it was a squeeze by CO or button, as that range is super-strong as well). In position vs a 3-bet, you want to do a lot more calling, and your 4-bet range would be fairly small. AA would be in a 4bet range IP, along with some other Ax hands that block AA, but that would be losing as calls. Depending on the situation, the 4-bet bluffs might be hands like AJo/AQo, A5o or A9-6s or A5-A2s. Just think of the worst Ax you'd flat with, and then 4-bet bluff with the next couple of combos down the list. (But if you're only 4-betting 6 value combos of AA, you only really need to pick one or maybe 2 hands as bluffs).
The concepts that are important in the spot you posted are:
a) Villain's squeezing range is tight/strong.
b) His 5-betting range is super-strong (KK+, maybe AKs).
c) Your 4-betting range should be polarized to hands that 4-bet/call (AA) or 4b and have easy folds vs a jam (Ax).

So QQ makes a lot of sense as just a call in position vs the squeeze. Since you hate getting shoved on when you 4-bet, and you rarely get called by worse, it's best to not 4-bet in the first place.

Thanks. This is very helpful.