Every time I do heavy squats I notice an extreme soreness and tenderness in my arches. Do you think the heavy squatting is responsible for this problem? What exercises can I do to strengthen my arches?
The first thing you might try is avoiding doing your heavy squats with your toes pointed out too far, and when rising out of a squat position don't permit the knees to buckle inward, because this position throws most of the stress or weight on the inside of the foot, and consequently the arches.
The feet should be laterally rotated but only slightly. Your best foot angle can be determined by hanging from a pull up bar. Observe the natural position of the feet. This is the foot angle to use for your squatting and deadlifting as well.
One of the best ways to keep the knees from buckling in is to develop a stronger set of abductor muscles (the muscles on the outside of the thigh). Also on your really heavy sets of squats use knee wraps. The function of these is to keep the knees in a straight back and forward direction and keeps the patella tendon from deviating medially or laterally and stabilizes the knee joint.
As far as strengthening the arches, include a series of calf exercises, standing or seated. These types of exercises influence the arches and serve to strengthen them, particularly if the feet are turned in, "pigeon toe" fashion.
Exercise the toes frequently by trying to pick up small objects, such as marbles, clothes-pins, etc.
If your condition of sore arches persists it would be advisable to visit a foot specialist to determine how severe the condition is, but in many cases the above hints will do much to improve it unless they are too far gone. Eventually you may have to wear arch supports as a means of relieving the pressure, especially if your job requires you to stand on your feet all day.
By all means be sure to include the practice of exercising your toes every day, and when the soreness is present, run cold water over your feet. Do this after your exercise session, too.