I have been waiting for some of the experts to chime in here, but am disappointed no one has.
My personal opinion is that box squat do not hit your quads very hard in general. I think if you set the box low and just tap it and go, then maybe, but if you set it at parallel and hunt around for it with your butt, then it's all posterior chain and back, very little quad except some of the drive part where your quads are not really being challenged.
I would love to hear some other opinions though.
I am Anton Zdravko Martin!
Best meet lifts: Sq 150 Kg (330 lb), Bench 120 Kg (264), DL 160 (352) @89 Kg (197)
Best gym lifts: Sq 375, Bench 280 (pause), DL 385 @205 or less
Goals beyond my wildest dreams, 600 Kg raw total, 200/160/240
Hack away at anything which isn't essential. Do what you love, and do it often. Fazc.
Everything competes for recovery so more assistance is not always the best idea. miked96