Where the hamstring connects at the hip must contract at the top of the squat. However, it is a secondary mover in any lift that is a “push”, such as squat, deadlift, and leg press.

You are right in saying that the glutes are primarily involved - but the quads are as well.

The quads are universally understood (and rightly so) as the prime mover in the squat because they do most of the work in a squat.

In this movement, the first thing that SHOULD happen is the hips move back. Then a bend occurs at the hips and the knees. At the bottom of the squat, the glutes and quads are stretched, while the hamstring is contracted at the knee (which eliminates it from being the prime mover because it is contracted when the most active muscles in this lift are stretched). In order to move from the bottom of the squat to the top, the quads and glutes MUST contract (shorten). As the quads contract, they pull the knees from a bent to straightened position, and the quads (with some help from the top of the hamstring) pulls the hips from a bent to straightened position. This is how the squat works, which is why the quad and glute are the prime movers.

Hope this helps and makes sense. Feel free to reply if you have further questions. Thanks for your post.