Can't kick on my back

No idea. How about on your side? Can you flutter kick on either side?

I had not heard kicking on your back was good for form (which only means I haven't heard it. I'm not an expert). I use it as recovery - I can breathe the whole way. Another bonus is that face down with a kickboard puts stress/curve on your lower back. Kicking on your back you engage your core and it's less back stress.

It actually requires good body position and quite a bit of core and upper leg strength to maintain a kick on the back. It helps to extend your arms fully behind you, streamline (google it if not clear what this means) and keep your head back o your hairline's in the water. If that still doesn't work, try doing it with fins; should make it much easier. I can't imagine why your calves and arches would cramp doing backstroke kick. Maybe you're trying to point your toes? You shouldn't be. In my mind (not a coach, just sitting on the couch trying to simulate it) the kick comes from core and hips, not lower legs or feet.

I'm sure the body position has something to do with it-- on my back my feet drop down something fierce. I'm surprised you say the feet shouldn't be pointed?

You probably drop your butt and your feet are flailing like mad causing you to cramp up ?

Make sure your head is back in the water. When you pick your head up your butt tends to drop down.

What other said on form but normally cramps are caused by weak or tired muscles. It's almost never low sodium like people think. We have tons of sodium in our cell. Anyway... I'd try to do some weight training once/week. I used to get calf cramps on long or hard bike workouts. They killed. I remember literally being stuck over my bike for 5 minutes on the trainer and I couldn't get off of it to stretch it out. I started doing calf lifts at the gym and the problem went away. Try that and see if it helps. Also make sure when you're doing these kicks that you didn't spend the past 30 minutes pushing off the wall like a mad man. That used to cause me to cramp up too.

Hands in a streamline or by your side? If you've done it one way, try the other.