Depending on form, the regular squat can definitely hit the quads, too. In fact, because my form was a bit off (I wasn't on my heels as much as I should've been), my squats were very quad dominant. The result is that my quads grew. But a few months back, I started to notice some pain in my left knee. Right below my knee, actually. With the help of a doctor friend, I narrowed it down to my peroneus longus, one of the muscles that runs along the shin. I videotaped my squats so I could show a friend who's knowledgeable about these things, and he pointed out that my left knee caves in on the concentric portion of the squat. That was the cause of my pain, but what was the cause of the knee cave? He pointed out that it was a lack of hip mobility and strength, as well as weakness in the adductors and glutes. So now, in an effort to fix all this stuff, I'm doing some very painful hip stretches that I obviously need to do, X-band walks, super light box squats, do squats with next to no added weight with a band around my knees to physically cue "knees out," and just focus on form in general. I'm starting to "feel" the correct form, which is the point, so I hope my squats improve when I get back to doing them.
I'm really, really looking forward to putting actual weight on my back again, but I need to fix this stuff now before it gets worse. Sorry for the ramble, but maybe some of you will find it helpful. Form, form, form!