I usually squat barefooted and at time with sandals. Get a better feel and grip with the toes, but than again I also surf (since I was a little kid) so going barefooted comes natural with me. Foot/toe gripping can be important in squatting. Should get a better flex/stretch to the movement too. My regular workouts are barefooted or sandals or flip-flops.
Some guy's ware martial arts slippers or wrestler shoes when doing leg work. A good friend of mine started wearing his old army Airborne boots and say they are the best for serious squatting. Olympic shoes have raised heels for better leverage/balance when squatting. Seen some guy's wear regular work boots. All depends what you feel best doing.
Just to note; some wear shoes when doing calf exercises, which can limit the full effect of that exercise. Try barefooted calf raises for that better feel , grip and stretch to the movement. Can help the front of the calf also. If doing a set of calf raises, at the end of the set hold the position of a full stretch (bottom) between those sets as you pause for the next set. In other words, don't leave the block, machine, etc between sets......just hold the stretch. Some successful BB'ers have suggested taking 30 to 45 seconds break between each set of any calf exercise. Good Luck.