Bit of a terminology quibble: “Shoving” refers to going all-in. ZMAN didn’t shove pre-flop - he “opened” (made an initial pre-flop raise), and then 4-bet to 10BB over your 6BB 3-bet. Shoving can be fairly profitable, but you definitely have to choose your spots. It’s generally better to make a shove than to call one, with some exceptions. Bounty tournaments where you cover the other player’s all-in comes to mind - due to ICM, if you’re in the very early stages of a tournament, but you have way more chips than the all-in player, it can even be profitable to call an opponent’s aces with 7-2 offsuit.
Back to the hand… quite a bit to unpack here. Trying to squeeze with Q9o is probably a -EV play, since you’ll likely only get called or re-raised by better hands, and you’ll be out of position for the rest of the hand. As @love2eattacos mentioned, if you are going to squeeze, you need to pick a larger size, since the pot has now ballooned to 9BB, and again, you’re out of position. Going for a 12BB 3-bet, 6x the original raise size, makes more sense. Best-case scenario, everyone else folds, and you’ve scooped a healthy pot.
Calling ZMAN’s min 4-bet makes sense, though if he’d gone bigger himself - as he should have with such a premium holding, facing so many people behind him who’d called his open - I would have definitely sent Q9o into the muck in the big blind.
Once you reach the flop, I’m not loving either the decision to lead out as the big blind, or the size. Let’s start with the first part - the decision to lead. If you’re betting only for value, people can “profitably” fold when you bet, so you need to choose some bluffs to work into your range. On a QQ2 rainbow board, in which you 3-bet and called a 4-bet, what bluffs could you have? AKs with a backdoor flush draw comes to mind - you’d be blocking both overpairs and have the potential to improve on later streets. However, there are only three combos of AKs with backdoor flush draws, and those will become tough to play on later streets if you get called and the turn bricks off, giving you neither flush nor straight draws. What other hands with low showdown and high equity might make a 3-bet and call a 4-bet - AJs? ATs? Both would be drawing very thin to KK or AA.
Because it’s tough to construct a flop lead range in this spot, standard practice is to check to the last preflop raiser - in this case, ZMAN. That would allow him (and maybe even tatt2b) to commit chips to the pot, which you could then check-raise, and possibly induce a fold or two.
Also, consider that with your exact hand, there’s only one other queen out there, making it unlikely your competitors have one. What hands worse than yours do you expect to call your pot-sized bet? AA and KK? You’re drawing very thin to the hands that would call and have you beat - tatt2b might have 22, and either tatt2b or ZMAN could have AQ - and if your opponents fold off hands weaker than yours, then you’ve missed out on a lot of value.
On paired rainbow boards like this, if I’m going to raise out of position, I’ll do it with a wide range of holdings, but I’ll choose a very small size, like 1/4 pot. That will allow me to get some folds when I miss - A4s, A5s, AK, 99, TT, JJ could all be in my squeeze-and-call-a-4-bet range that missed this flop - and value from my opponents’ hands like KK or AA when I have AQ, KQ or QJs and make a set. If I bet with my missed hands and get re-raised, I don’t feel so bad about letting it go, since I’ll still have a healthy chunk of my stack behind.
Does it suck when you’re way ahead after the flop, all the chips are in the middle, and someone else sucks out on you? Absolutely. However, you probably have a bit less equity in this spot than you think. tatt2b could have caught a 9, or made a backdoor flush. ZMAN could catch an ace. Plugging this into an odds calculator, at the flop you only have a 60% chance of winning outright, and a 17% chance of chopping with tatt2b. tatt2b had a 15% chance of winning outright, and ZMAN lagged with a 9% chance. Those are still odds I’d take all day long for the chance of tripling up early in a tournament, but if you only have a 60% chance of scooping the pot, 40% of the time you won’t. Being able to handle that other 40% of the time - whether mentally, or with your bankroll - is key to being successful at poker in the long run.