You are thinking of it all wrong. It's not your fault, everyone does it. Get away from thinking of the lift in terms of quads, hamstrings, glutes, and differentiating between the muscles involved. If you watch a really, really good squatter, regardless of geared, raw, foot position, amount of weight, they all do the exact same thing.
They abduct their hips (hips open up and knees go out), hips flex back, trunk flexes slightly to keep the weight over the mid foot/heel, and then it is an abduction and extension to get out of the hole. Just make sure you are doing the movement correctly and you will be fine.
As far as different muscles hurting more, squatting sucks. And it doesnt feel very good. So don't worry about it. Definietly start doing some box squats.