It depends on whether you're doing low bar or high bar squats. 

With low bar squats, you're supposed to be keeping your chest down and leading with your hips as you rise, so what your description would indicate to me is that you're picking your chest up too early. Pulling down on the bar shouldn't be necessary, because the bar rests on the shelf you make with your delts securely enough that (ideally) the only function of your hands is the keep the bar from sliding down your back. Your arms are locked in really tightly, so the focus is keeping your shoulders tight, rather than pulling on the bar. Some people can even squat hands-free in this position (as a form exercise-- not recommended at training weights). 

On the other hand, with high bar squats, pulling down on the bar like you're saying is the standard advice. It does help keep the bar secure, but more importantly, it engages your lats and helps keep your core tight. It also helps you remember to keep your elbows down, thus your chest up, which is important for high bar squats.

You should probably pick low bar or high bar rather than using a mixture, because the mechanics are actually pretty different, not just from pulling the bar, but throughout the whole movement.