I've also watched your high-bar squat and noticed that you seem to have a habit of lunging your torso forward at the bottom. This is either due to weak flexibility, bad technique, or a weakness of your lower back.
If it's a flexibility issue, it's probably your hamstrings, hip flexors, or groin that is keeping you from squatting properly.
If it's bad technique, you have to tighten your core throughout the entire lift. TAKE AS MUCH AIR AS YOU CAN BEFORE YOU EVEN DESCEND AND KEEP ALL YOUR CORE MUSCLES FLEXED THROUGHOUT THE LIFT.
If it's a weakness of your lower back, you need a stronger lower back.
I think the best thing for someone who wants to learn how to full-squat properly is to learn how to do an overhead squat with a stance slightly wider than shoulder width, full range of motion, and without losing your balance the entire time. If you can overhead squat properly, you can back squat properly.