There is really no way to know if it will carry over or not. My guess is that it won't. The box squat tends to take away the ability to rebound out of the hole for raw squatters in my experience. You have a safety net in the box being there, so your form/mind set changes to reflect that safety, which doesn't help for the regular squat.
Also, if you are squatting correctly, your hams and glutes are likely getting enough work already.
There is nothing wrong with giving this idea a go, though. Just make sure that you are giving maximum effort and dedication to the main work sets.