Really need to address the upper back and the elbows cranking up. Your upper back is rounding on the way up for most of these and it's playing havoc with your ability to lean over and stay balanced throughout the squat. Lift the chest and pull the elbows down at the top before every rep. Also, put some athletic tape around the middle of that bar so it's less likely to slide around on you. If you can learn to keep the upper back tight for each rep, you'll be able to address the balance issue and smashing into your knees in the bottom.
Lifting shoes would also help rather than boxing shoes.