The squat is a different exercise than the leg press because it incorporates more stabilizers into the movement and mimics real-life motion much more closely than an assisted leg press. The movement involves gravity as the knees bend and the upper legs lower into a squat. Gravity forces the quadriceps to use the upper body as another free weight, one whose shape and size is exactly what nature intended it to lift. The hip adductors and abductors are working together to keep the knees over the line of the feet. The lower back and abdominal muscles are contracting to keep the upper body strong and straight.
Good posture requires the entire body to work together.
And while all this is going on, coordination and balance is developing as the body tells the brain via the nervous system where it is in space. To all goodness there is an equal opposite, however. Free weights require practice and maybe assistance from a professional. Try the exercise first without weight or very little so the body learns the motion. One should be moderately strong and healthy before picking up much weight.