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Daniel, the hip joint is in a flexed and externally rotated position in the bottom of a squat, so primarily those things. Abduction is also coming into play when you shove those knees out. As far as actually working on the hip in a muscly sense, if you are staying engaged and actively attending to all the stability cues (actively locking your feet into the floor, shoving the knees out, loading the hamstrings), then you will also improve your ability to perform these cues and gain some endurance in holding a good bottom squat position.
You could also bias that deep squat position towards different mobility elements by changing your stance and position. For example, squatting with a wider stance (still getting the knees above the feet) will help you open up more abduction, while squatting with a narrow stance places more demand on the ankles.
Dorian, as Conor said it’s more important that you are able to stabilize the foot and lower body. Squat in a way that will allow you to get into the best overall position for what you’re working on. Having your feet straight in the squat might be a good goal to shoot for, as far as squat mobility is concerned, but trying to go too parallel before you’re ready will probably just impede your ability to put some time into that squat position, and might compromise you in other ways. It’d be like trying to squat to full depth when you can’t. As your hip and ankle mobility improve, just keep adjusting your stance to reflect a closer approximation of the ideal stance.
To the original question of the thread, db2000, spending some time daily in the deep squat (similar to the 10min squat stance) is a great way to improve that position. I’m glad svinto posted Ido’s squat routine, as it will help you direct more focus to specific corners of the squat while you’re down there. You might even consider starting off with a shorter duration, and adding minutes over time, like how you might progress other elements of fitness.
Starrett mentioned in a Daily Rx recently about taking the 10min squat test in chunks, and adding mobility work in between, which is (obviously) a great idea. He was talking specifically about lower leg clean-up, but you could use that squat test as a more general diagnostic too, of course. Spend 2min in the deep squat, note what feels sticky or where you’re lacking mobility, get after it for a couple minutes, get back in the squat for another 2min. Repeat as desired!