I consider myself pretty active at 40. But for the last 10 years or so, I have fought increasingly severe wrist injuries. I can injure it doing something as simple as getting up off the concrete floor in the shop. The pain lasts for weeks at least, and is always easy to re-injure. Both wrists have this trouble. I fell and caught myself with my right hand this weekend, and of course that aggrivated the wrist. I wore a brace for the rest of the day as we gathered firewood. Now I have a lump with no color on it about 2" up from the wrist that is tender. It is around 1/4" high. Particularly when I have injured them, they are easy to pop at the joint.
I use my hands in everything I do, and my wrists are really starting to hold me back. Working out is nearly impossible now at least wtih weights. I visited with my GP, and he suggested ibuprofin and squeeze balls to strengthen my forearms. I'm not sure that those things have helped long term.
If you are injuring your wrist because they hyperextend, then strengthening the forearm muscles is about the only way to protect them. It is the muscles that will keep the wrists from extending too far. When the wrist hyperextends, the extensor tendons can be caught and pinched, setting up a tenosynovitis within the extensor tendon compartments.
However, squeezing a rubber ball will not do too much. This is essentially an isometric exercise for the finger flexor muscles. You will need to strengthen the wrist flexors (FCR, FCU) and extensor (ECRL, ECRB, ECU). Strengthening the finger and thumb flexors and extensor will be of added support, but they are not primary. To build strength throughout the range of motion, you have to exercise throughout the range of motion.
A very good exercise for this, and you can do this at home, is to attach a weight to a rod (eg broom handle) with a rope. Holding the rod out in front, with the palms up, roll the weight up and down. Then, with the palms down, do the same. You can also go the the gym and do wrist curls, both for extension and flexion.
To strengthen the finger flexors, the use of therapeutic putty is the best, as you are able to contract the muscles throughout the full range of motion (pulling the fingers all the way into the palm). For finger extension, use of rubber bands actually works quite well.
You should probably avoid exercises that put significant weight on extended wrists, such as push-ups. When doing heavy weight lifting, you may need to taped your wrists or use gloves/splint that keep the wrists from hyperextending too much.
If however, after you have strengthened your forearm muscles and you continue to have problems, you should probably see an orthopedic surgeon or hand surgeon. If you have irritated the extensor tendons enough, and have set up a chronic inflammatory response, you may need further treatment.