Proprioception is the body's ability to understand where it is in space in relation to itself. Typically after an injury, proprioception decreases, especially if you have an ACL injury as an example. This is why the bosu ball is useful in rehabilitation for reestablishing lost proprioception through unstable surfaces as the GTO's (golgi tendon organs) will be able to sense the force/tension within the muscle and allow for adjustments.
http://courses.washington.edu/conj/bess/spindle/proprioceptors.html
However, in healthy individuals or those who are post rehab (done with their rehab program) this is unnecessary largely and one should continue with barbell squats as able.