Ask any race horse jockey about using your knees. When we squeeze our bike, its because of your knees. Jockyes will tell you the same thing.
In reality, we usually have our knees above where the contact is made, but it's your knees doing the work.
I have to disagree with you on the push out only thing, Kawie. When we squeeze, call it knees, calves, legs, whatever....we take pressure and weight from our upper body, which actually helps to make us looser and more relaxed. Also helps to loosen our grips on the bars.
Try riding no handed while standing, first with just pushing out on the pegs, then try it (after youve gotten back up and caught your breath) while squeezing! :D