I am waffling on this. I think I could be convinced by any strategy that tries to get all of UTG+2's stack in the middle, and I could also be convinced to fold to a shove by UTG+2 if you call or min-raise the flop. I think this really comes down to reads you might have on UTG+2. If he's tight, then I'd be inclined to fold against a flop reshove from him. Otherwise I'd feel pretty good about my chances.
What do you make of UTG+2's 4x open followed by the small flop bet into four other players? I think most players with an overpair might bet more on the flop here, and I think the same for players flopping a set in an attempt to charge more against the flush and straight draws. But then some people are largely unaware of the pot size in live games where you have to keep track by yourself. He might think it's a strong bet since it's more than twice what he bet pre-flop. Either way, you have a better than 2:1 edge against even a 10% opening range.
The all-in player probably has a strong hand, since he was willing to shove into an open bet with three people still behind, and he's got essentially zero fold equity against UTG+2 and not much more against the other players. He might even have you beat now, but your equity against even the best hands on this board is quite good with the flush and back-door straight draws. He probably would have raised preflop with AQ, QQ, or TT, so he might have a lot of AcXc, KQ, QT, or 44. You're pretty close to even against that range, and if you throw in any paired tens like AT or KT or worse queens for him, your chances get lots better.
Clearly you don't want to fold, and any more action in this soon-to-be-bloated pot is probably going to commit your entire stack. I don't think that's a bad thing, though. You might shove, and I think that'd be ok with the stack sizes. I'd at least be tempted to call here to try to get more action from UTG+2, though. UTG is likely to fold to the substantial action in front of him since he's shown weakness so far, and UTG+2 is the only one likely to pay you off if you hit.
If you call, I think the biggest question is what UTG+2's reraise range is, whether a min-reraise or a shove, which are more or less his two raising options with this low SPR. If he's a player who wouldn't reraise here at all into a dry side pot without having you beat (for example if you could be confident he had AA, KK, AQ, or KQ), you might reasonably fold to a shove, but I think it's close. If he's likely to keep playing with all of his ~10% pre-flop raising range, I'd probably call. I guess I don't see a lot of argument for reshoving on the flop since he isn't likely to be raising with a bluff into a dry side pot given your range and the stack sizes, and he wouldn't call your shove with much you're ahead of. On the other hand, you have an SPR of about 1, so you don't really have a lot of options, so shoving when you have good equity won't be terrible. I'm not sure there's a lot of difference in your own range between calling and min-raising the flop all-in, given the low SPR. You could have AQ, KQ, QQ, QJ, QT, TT, 44. You probably don't call the flop all-in with just clubs or a paired ten, and probably not even QJ unless you have clubs, especially with UTG+2 yet to act. QcJc might be the worst hand you ever have here, so if UTG+2's willing to play back at you he's pretty strong.
I'd probably check my entire range to any turn card. If UTG+2 shoves any card that's not an A or K, I probably call. If he shoves a non-club A or K turn, I'd probably fold but not be happy about it. If he bets small, I'd probably float the turn and re-evaluate the river. My rationale here is that he's shown strength the whole hand even in the face of my showing strength, and much of my equity against him relies on the flush coming in. I don't think he's ever folding a better hand than mine here, and if the flush doesn't come, I don't think I'm a favorite, even given the pot odds. Also, the all-in player isn't really dead money, since he's likely to have gotten a good piece of the flop, so that minimizes the value of any perceived overlay from the main pot.
So, I guess my attempt at an answer basically ends in a shrug. Get it all in on the flop, or not.