Guys, comparing a full high-bar Olympic-style squat to a parallel PL squat is like comparing apples to oranges. Different styles will work your muscles differently.
Having said that, I think that, in general, high bar full squats are better for athletics. They tend to work your musculature more "evenly", whereas the PL-style parallel squat works more of your posterior chain. Plus full squats have the added benefit of giving you strength in an extreme ROM, which means less injury potential later on. I do both and like them both, I just concentrate on one or the other more depending on what my goals are at the time.
As far as box squats go, they're great for teaching you how to explode out of the hole. The box disengages the hips when you sit down and you have to re-engage them to get off of the box. I don't know if they'd be the best choice for hypertrophy, though.