It's actually the angle of the torso to the ground that is going to most strongly determine how hard the glutes are working during squats than the width of the feet (though, obviously you aren't going to be doing a very hip dominant squat with a very narrow stance or very wide stance).
Whenever looking at resistance training exercises you must take leverage and ROM into account. Since the glutes extend, hyperextended, abduct, horizontally abduct, and externally rotate the hip, you have to look at how much each of these actions is occurring during any given squatting exercise, and at the leverages experienced in any of them as well.
So, for instance in a front squat even though if you go ATG you will technically get maximal hip flexion and hip extension, and you may be isometrically horizontally abducting to maintain hip/knee stability (activating the glutes in the process), because your torso stays very upright during the motion the leverages (and therefore forces experienced) by the glutes are somewhat minimized.
A low bar back squat where you hinge primarily at the hips (and thus your degree of torso forwards lean is much greater than in the front squat), or a Zercher squat (where the resistance is always in front of the center of gravity and therefore the lever arm is again relatively long) on the other hand will require much more force to be overcome/produced by the hip extensors.
Going very wide does maximize the degree of horizontal hip abduction, but also makes hinging forwards/hip flexion from the hips challenging unless you have great hip mobility. But it does recruit more glutes than a narrow stance very upright squat.