Your feet possess TONS of sensory organs and receptors along with lots of muscles. You should be able to grip the floor, push out to the sides of your stance, and drive all the force that you can through that floor. This is HUGE for not only force production/strength/power/stability but also for postural positioning the rest of the way up the body. If you are trying to squat in an average pair of running shoes, then you are not going to have the squat form, strength or stability that you are looking for. If you are unstable at the floor, you better believe that you are going to be unstable at the knee, hip, trunk and the rest of the way up. I made this mistake for years when I was younger and suffered plenty of less than desirable squatting and deadlifting workouts than I should have. To quote my friend and colleague Therese Miller from an article:
"For thousands of years we roamed the forests and our feet were forced to grip and bend with every step. Most of us have been in shoes since we began walking. Our feet only grip a flat, solid surface and rarely if ever bend because of the rigid soles. Many people know they "overpronate" and therefore are told they need a specific shoe to correct it. We rely on "arch support" or on stabilizers to prevent our ankles from rolling. The result is weakness, stiffness, and often injury."
Since most people are stuck in ridiculously designed shoes most of their lives, the foot muscles never develop how they really should and problems come about from the heel on up. The foot muscles need to be developed just like all of the other muscles people typically train.
Shoes such as Chuck Taylors, New Balance Minimus, Vibrams or even barefoot are the best way to go when looking for good shoes to train in. You should be able to grip the floor, flex, extend, and move your foot and toes every which way with the shoes you have on. Not only does this work for training, but if more people would get out of heels and clodhoppers throughout the day and wear shoes that were closer to barefoot, issues all the way up the body including back and knee pain would be a lot less problematic. When a girl wears high heels (which I'm not saying I don't like, they're just terrible for posture :) ), the ankles get stuck in plantarflexion, which in turn causes the pelvis to go into anterior tilt, which in turn causes the lumbar spine to hyper extend, which can cause the thoracic spine to hyperflex, which in turn can cause the cervical spine to hyper extend, which all leads to unwanted stress at the knees, back, ankles and neck. What happens at the floor carries up the rest of the body. Develop the intrinsic and extrinsic foot muscles, fix your posture from the floor up and notice huge changes in your entire kinetic chain in both feeling and performance.
On a side note, when it comes to running (and even walking for some), minimalist shoes can be good; however, you need to gradually work your way to that point and build up the tissues tolerance in order to handle that kind of stress through the foot and ankle without problems. This is especially true for runners who have been used to wearing shoes with a higher drop and extra cushion.
Now go get some Chuck's and get to the squat rack!!
And definitely check out Therese's blog series about foot function and shoewear http://millerswc.com/blog.html She has an amazing article series throughout this page so definitely check it out. She knows more about foot mechanics than anyone I have ever met so go learn something.