I have been having similar issues in the squat, and have worked/am working a few out with the help of others.
As already said, work your hamstrings and glutes. Glute-Ham Raises are not a bad idea and are easy enough to do in most gyms. Also, just really think about forcing your knees out every rep, especially on days when you might do high volume work and will be able to "grease the groove". I don't know your warm up routine, but also maybe your hip flexors and hamstrings are just tight so try stretching/foam rolling them well before squatting and do some light (~ 30 lbs x 2)goblet squats to stretch them more.
From the video, its tough to tell if you really set a solid base before squatting. Try walking out in 3 steps: left foot straight back, then right foot back and to the side, then left foot out to the side (or start with your right foot if you prefer). Once you've done that, be sure to pinch your glutes and fully lock out the weight, then take a second to let the weight settle on your back. Once you've walked out, locked the weight in a stable position and settled it; begin to squat. That alone made a huge improvement on my knees caving because I felt solid from the start and I saved energy on the walkout.
Sorry for the long post, hope some of this helps.