If you do an olympic back squat (bar placed high, narrow-moderate stance, ass to the floor) you'll get a lot of quad work plus the posterior chain.
I don't think very wide stance squats and box squats are a particularly good movement for functional quad strength, and (this is just my opinion) the ultra-wide stance glorified good morning, also known as the modern day PL-Squat is very taxing to the posterior chain, but does nothing for the quads and the only thing it has carry-over to is being a good ultra wide squatter.
I think your original question is confusion about the terms 'back' and 'front' squat.....stop thinking bodyparts, it is all about movments. Back and front just refers to bar placement during the lift....racked across the front or on the traps.
Everyone was pretty much dead on about what lift stresses what.