You literally cannot perform a squatting motion without the quads engaging. Just because you don't "feel" them does not mean they aren't working.
Don't use where you feel a movement to determine what it's actually working. Use the actual joint movement and moment arms to determine what it is actually working the most.
Here are some examples:
http://thefitcoach.files.wordpress.com/2012/06/squat_bar_placement.jpg
The activation from quads, to glutes, hamstrings, erectors, etc, depend on the type and depth of squat you are performing as well as your own limb segment lengths.
There is no reason to do any isolation work such as leg extensions or leg curls unless you are a body builder or are rehabbing an injury.
I like cross arm front squats for runners and other non-strength focused athletes because it's easy to learn, easy to execute, and there's less stress on the lumbar. It's easier to poorly execute a back squat and still get the rep up with poor form/"stripper squat". If you perform the front squat poorly/chest cave in/lean too forward....the bar will roll off your shoulders.
https://www.muscleandfitness.com/exercise/workouts/leg-exercises/crossed-arm-barbell-front-squat/
Alan