Ouch. So you're saying that you shouldn't wear shoes while doing squats? Never heard of that, would be nice to know...
Discussion
Yea my trainer suggested being barefoot for squats as well.
There are shoes that are suitable for squats and deadlifts. Running shoes are definitely not.
Barefoot is always better than with running shoes.
The problem is the soft sole, which constantly corrects the center of gravity at the smallest loads and then misloads your joints.