Yes you can definitely use the bellicon with bare feet or socks with rubber grips.
Do not use regular socks as you may slide off. You can also use shoes, however it is a great workout for the muscles in your feet if you go barefoot. If you normally wear inserts in your shoes for everyday walking, you may want to use them while bouncing as well.
Here you can see our Akkua PVC socks
Here you can see our Akkua Pilates socks