Besides flexibility, working at it, and general strenght and stabibility...the secret to your squat "problems" are all in the glutes. On your squat vid I noticed that your lower back rounded out at the bottom a bit and that you just said you are not going as deep. Also that you are working on keeping your knees out..which I take to mean keeping them from caving in rather than having them stick out
Try these two tests if you can. Get down into a squat position, bw only, and take care to really try to tighten up and engage the glutes. If you can really get them firing what you will notice, in the bottom position, is that as you tighthen them your knees pull outwards. All the bullshit little methods people prescribe, like putting a ball between your knees and all that, are just that...bullshit. It's all in the glutes, baby!
Same thing for the back rounding out. For that try this exercise/stretch for working on depth without the butt-wink (I really shouldn't be giving this secret away
).
Stand in front of your rack upright so you can hold on to it. Grab onto it at around waist height or so. Get yourself into a position for squatting. Now what you want to do, rather than just sitting back and dropping into a squat is to PULL yourself down into a squat using the hip flexors. I'll expand on that.
Just like that little relationship between the knees and the glutes there is a relationship between the hip flexors and glutes. What you shoud notice is that when you tighten and engage the hip flexors (and quads) the glutes should fire and tighten up also. If they don't, then you have discovered a fundamental problem...your glutes ain't cominng to work so you would have to address that.
So, after establishing a good tight, neutral spine, you pull youself down into a squat using your hip flexors. Like pretending they are rubber bands. As you do that your glutes will engage and feel very tight and active along with your hams to some extent. As you get into the bottom position you want to keep the glutes firing. Then you can try to establish more depth. Sort of push down and wiggle into it. All with the glutes working. Here is the other little magical relationship you should notice. When your glutes are really working in the bottom position like that, your lower back won't cave under. It can't. As soon as you relax that relationship while trying to get deeper, the back will begin rounding out more.
That is the secret to it. It doesn't mean that it won't take work to get this going while you are actually squatting. But it gives you something to go far that is a lot less nebulous than this whole working on it though will alone. If you do try box squats this is something you need to be aware of. Most people actually completely relax their lower spine when they sit down on the box.
Things to engage the glutes more wouldn't hurt. Glute bridges. I find pullthroughs to be fantastic for that also.