This is where a lot of confusion can come in. Box Squats have a very broad perception.
There is Westside with Louie Simmons, who talk about Box Squats being more like a leg curl because your shins are perpendicular or beyond and you must sit back on the box. This will incorporate more of the hips, glutes, and hamstrings especially when come out of the whole.
Then there are those who simply use box squat as another variation to make sure they are hitting correct depth and may also take the idea from westside to break up the eccentric and concentric.
There different ways one can use the "box squat" and it depends on the training method.
Here are my views when it comes to quad activation:
I feel using a narrower stance, hip width apart, hits my quads much more. Also, using a higher bar position on the back and a more olympic style squat helps this out as well. You can use these ideas with the box squat.
I've done box squats with a hip width stance high bar position and it smoked my quads. I wasn't trying to sit far back on the box, although i do incorporate them sometimes. My main goal was to hit depth and break up the eccentric/concentric.
The term "box squat" has different views from different schools.