Try goblet squats first? To help activate the glutes twist your feet outwards without actually moving them. Try to build up that mind muscle connection I suppose. From what I can remember you have to pretend there's a ball between your knees and you need to keep it in between them the entire movement. There's probably a comrade who can go into more detail and explain better. Otherwise do hip thrusts or donkey kicks for glutes, or the 'sus' machines (abductor and adductor)

try holding a plate to your chest and doing it that way and see if it does, probably form related

People keep bringing up form but high bar back squats are pretty fucking quad dominant. If you're doing those correctly it's going to feel like your glutes aren't taking over because they're not supposed to be. Low bar squats encourage more posterior chain activation but that bent over shit with heavy weights seems sketchy to me.

IMO if you want to specifically blast your ass just do Bulgarian split squats.

Edit: Late morning reading comprehension came through; front squats are still a quad dominant movement though.

+1 on the form guess. if you're going to a gym and they have trainers maybe you can free-roll a couple questions

you're probably just extremely hamstring dominant. Get a resistance band, string it from one ankle to another so there's a little tension in your gluteus medius (outer part of your cheeks) - you can do some side-steps with this to warm up at this point. Now squat with that tension and some weight, go up slowly weight-wise and just focus on your glutes firing with every rep. You'll have to do some re-conditioning for your nervous system to re-learn the movement, but it's worth it for a stronger squat (and a bigger butt, eventually)

as to why it works fine without weight: I'm guessing you're 'jumping' more without the axial loading, since it's easier to do so. Do include some box-jumps or something like that, if you're up for it. It's fun if you have the ceiling height for it

Another possibility is that your glutes could be overly (and perhaps chronically) tight, which was my biggest issue with squats and general glute activation.

When your glutes are overly tight, they’re effectively “sleepy”, so when under load, your glutes are like “shit man, that’s like… a lot to like, ~do~ right now soooooooo… we’re gonna go take a nap.” Their more “awake” homies, the hams and quads, will allow your glutes to continue to “sleep”, thereby protecting them (and thus, you) from injury. If this is the case for you, you just gotta wake those babies up with some muscle stimulants, and stretching is like amphetamines for muscles.

For me, the key was lots of pigeon pose and figure four variations and an obscene amount of rolling (maybe too much??). Before workouts, after workouts, midday, midnight, literally whenever. It hurt like heck and made me feel like a stupid inflexible oaf for the first couple weeks, but so so soooooo worth it in the long run!