You may have sore feet from over-pronation. If it's from the squats it could come from a lot of lower body problems but what tends to happen is that people over-pronate to make up for a lack of range of motion. So basically you transfer more of the weight to the inside of the foot than is healthy. This can actually cause the feet to flatten as in "fallen arches". And you can get pain on different parts of the bottom of the feet.
If your knock kneed or your knees turn inward this would be a clue and you may tend to have that happen during squats.
Typically with shoes for squatting and deadlifting they have flat soles. Lot's of people like to squat in chuck taylors and they offer absolutely no support to the bottom of the foot. Not you may be a person who need some extra support but the point is that squatting in itself should not automatically necessitate this.
Maybe you should get it checked out by an orthodedists or something like that.
But I think some shoes are a good idea in general
