Primal Movement Pattern
Our ancestors used to perform many daily functions (ie hunting, cooking, eating) in a full squat. We've lost the squat to sitting in western culture, but you can still see it elsewhere in the world and always in children - it's one of our natural positions.
I often hear "I can't squat, I have bad knees." This is exactly WHY you should! Bending and stretching of the ankles, knees, and hips allows more oxygen to flow around the joints, improving flexibility and building strength. This stretching also gets the blood pumping throughout the entire body which allows more nutrients and oxygen to travel to vital organs and muscles. See a beginner level squat here.
Full Body Work-Out
Even though you are working primarily with your leg muscles, the movement creates an anabolic environment which makes the body release hormones vital to the growth of muscle tissue, promoting muscle growth throughout the entire body.
Other means of exercise (particularly running) can put strain on the knees, back, and ankles. Squats do not do this, and in fact can strengthen those areas.
Get the Junk Out
The muscular action of a squat improves the pumping of fluids in the body, which helps with constipation, and aides in detoxification through the removal of waste and delivery of nutrition to all tissues. It also naturally aligns the intestinal tract in such a way that the intestinal wastes are moved downwards. The squat is also the natural posture for defecation.