Wouldn't full squats activate quads more than gluts?
The glutes and hamstrings are recruited most when the weight is in the back and on the heels, while quads are recruited moreso when the weight is the ball of the feet and forward of the shoulders (front squat).

The Single Leg Squat is great if the person has balance, however, overall progression is limited. It is a great dynamic activity. A full squat on a leg press isolating the glutes with progressive overload would 'build' the glutes as per the goals suggested by the original post

Bridging builds strength, and is great for muscular activation, but beyond that most people will not develop enough overload to sufficiently increase size of this musculature.