Preflop: I squeeze for several reasons.
The player UTG raises wide from this position.
Player on the BU can call with a wide range knowing this, and he's flat called before (as loose as 63s and A3o in the blinds).
They both have a very high fold to 3bet% and I look strong here.
In the unlikely event that they do not fold, their call 3bet% is still higher than their 4bet%. This means that if we go postflop, I'm playing an unlikely holding that can flop well deceptively. In the rare event that I do make a straight, there are plenty of AQ / some AJ / the occasional AK in the average 3bet calling range that might stack off with the pair.

Flop:
Best case scenario. I flop trips.
I don't want to lead out against two players after squeezing because I may scare some 99/TT and I want to encourage AK / AQ to bet. If villains hold strong broadways it doesn't hurt to give them the opportunity to pair up either.
I check/call flop, representing 99-AA basically. I don't expect them to EVER have a 3, or to ever suspect that I have one, and I don't expect them to have 55 often at all (MAYBE BU). I don't need to raise to get it all-in easily on turns and rivers, and I want to encourage JJ/QQ to keep valuebetting.

Turn:
Very dynamic card. Anyone with 99-KK etc. has to be scared now. If I lead out on the turn I think I'll fold out these hands a lot, but if I shove a blank river (or even a broadway or something) they might level themselves into a call thinking I'm bluffing / getting stupid with QK on a K river etc.
Turn checks through.

River:
If I shove, I expect to get calls from AJ/AQ/AK always. Weaker aces maybe since I called turn. I expect to sometimes get calls from QQ/KK thinking I'm valuebetting a J or bluffing after the turn check through. JJ/AA are all I'm worried about, and AA would shove over the squeeze preflop a lot of the time.

tldr

3way pot, they're only calling with an A, so jam

seems too wide preflop
postflop bet bet bet
99 TT isn't folding flop AK AQ prolly either when you bet smallish just enough to bet bet shove comfortably and you lose value on overcards to pocket pairs

bet just enough each street to get stacks in by the river. no one is folding to bets post except axs types

build,that, pot, sir!