First of all, you said you have flat feet. Why would you need good arch support if you don't have an arch? Aside from that though, you want shoes with a flat sole for squats, not 'athletic shoes'. Converse Chuck Taylors, or shoes made for weightlifting are best.
How much weight are you squatting?
I don't necessarily believe bringing in your stance will improve the foot pain situation. I get the same pain, and I squat with a relatively narrow stance. Right about shoulder width. It's worth trying though.
I tend to think the pain comes from a weight shift forward in the squat. If you're able to maintain your balance on the midfoot, or towards the heel, the pain shouldn't really occur. I have a tendency to push off my toes a bit sometimes, and I think that results in foot pain.