I don't think he'll re-raise the UTG opener (+ a caller) here that often with a mediocre / bad hand in this spot that often as a squeeze play. If he's aware of position, he'll more likely re-raise a MP or late opener. An UTG raise usually represents genuine strength. After his 3-bet there's ~20BB in the pot. Calling would increase it to ~30BB and put you out of position. I'd 4-bet him to ~40/45BB. He'll likely 5-bet you with a monster. If I decide to flat with AK I'll make sure I'm in position, out of position I want to take the pot right there.
If I just flatted with AK out of position and missed I'd probably give up at this point. This flop probably missed him too, depending on his c-betting percentage / aggression I might float and try to take it from him on a later street. You can use a flush scare card to your advantage should it arrive, but min-raising him makes this less believable. Just calling would make it more likely. Don't think you can rep a straight here often.
Calling pre-flop and min-raising him here, he'll likely put you on a medium overpair 77 - TT, maybe even JJ. Most flush draws would probably just call, but always a chance you are semi-bluffing with one. Min-raise also comes accross kind of weak. You probably would have raised more with a strong hand (set of 6's for instance).
He just calls, if he has a set he'll most likely be holding 66. Would expect him to come over the top with it. Also would expect him to just call on the button with a small pair hoping to spike a set. For now I'm putting him on overpair TT+, or a strong flush draw A:clubs:K:clubs:, A:clubs:Q:clubs:....
That 4 likely did not help him or you. You don't mention any action on the turn and your shove on the river is equal to what you have left at this point (118.85BB), so I'm assuming it went check - check. Your checking basically announces that you were semi-bluffing and missed the flush. Most medium overpairs would have bet, but he can't rule it out yet. Maybe you were just intending to check-call for pot control fearing a bigger overpair.
The flush arrives. Slight overbet, but not that much. His range is mostly likely TT++, or a flush.
Let's assume he'll only 3-bet AK, AQ of clubs (if he has the flush). Other 3-bet hands might ofcourse include JJ+, AK / AQ of other suits...etc.
- 16 ways to be dealt AK and also 16 for AQ, totalling 32 hands. But ofcourse only two of those are AK / AQ of clubs. Leaving only 2 hands.
- 5 possible overpairs (TT++), 6 ways to be dealt each one. Resulting in 30 hands.
- He's not bluffing here IMHO.
That gives a total of 32 possible hands of which you are beating none. You can shove to represent the flush, he'll be hardpressed to call if he doesn't have it himself. But it's basically the only way you can win at this point. He'll most likely bet his flush, which you can't call and he'll check his overpair.
So his range consists out of overpairs for the most parts, flush is a smaller part.
I think you are getting stacked here most of the time. Your line comes across as weak. Call, call 3-bet, min-raised low flop, check, shove. Think he will call here a lot of times thinking his overpair beats yours or put you outright on a bluff.
Update: Some edits as I miscounted the AK / AQ combos.