A bit of a bump (Hi, Darksakul!), but just my 2 cents:
Square gates are usually best for fighting games, and they make DP’s (Dragon punch’s) very easy to do. The original street fighter 1 and street fighter 2 arcade machines that I played on had square gates and I loved them. I could do Dragon punches all day and almost never miss. The ones that had circle gates…well…I lost literally every match or tournament I played in because I could NOT DP on them! Whatsoever. The people that did the motion starting with diagonal, ->forward down diagonal could do it on circles, but there was a triangle method which allowed you to get very meaty DP’s, and that’s too difficult on a circle gate. I also find that difficult on octos. So yeah, I loved square gates for dragon punches.
However, square gates just completely suck for NON fighting games, like platformers or side scrollers (although games like Raiden II, where you are always moving diagonally might be different), or 8 way games where there is more emphasis on primary than diagonal directions. The old gauntlet games are an example. On a square gate, you will feel awkward and will wind up moving diagonally randomly, and it’s VERY hard not to ride the gate in games like these where mobs are thrown at you relentlessly.
These older arcade games are best if you ride the gate, and octagon gates just feel the best here!
For USF4 and street fighter 5, due to the importance of diagonals, and for ease of DP’s, definitely prefer the square gates.
If you like playing multiple arcade/console games and switch between them, you should buy all three gates (I mean, they’re cheap, you know?) and just swap between them depending on the game. It takes more time to unscrew a joystick bottom panel than it does to swap a gate.