If you are uncomfortable from sleeping position, you should be allowed to try a different position. All surgeons have a different post op protocol - I allow my patients to sleep in any position that they are comfortable in. Generally, the implants cannot be moved out of place by a funny sleeping position - the pockets are usually pretty well defined. However anything to the extreme can contradict generalizations - high impact or awkward positions repeatedly could potentially have some effect.
Consult with your surgeon and see what sleeping alternatives they suggest as well.
While every surgeon has their own recommendations for postoperative activities, including positioning. Most of the time, lower back pain after breast augmentation is due to constant pressure on one area of the skin and/or soft tissue, or muscle tightness. In either event, I have never felt the need to be so restrictive on my own patients' activities or positioning that we would create other problems in distant parts of the body. Typically, as long as you are wearing an bra which provides appropriate support to the breasts, you should be able to lay supine, or on your side, as soon as your own comfort allows. Be sure to check with your surgeon about positioning, and be sure that they understand the difficulty you are experiencing with your back as well. Also, you should try to discern what part of your back is hurting - is it the skin? Are you developing pressure points or pressure wounds on the skin? Or, is it more the muscles and joints? That will also help your surgeon to better manage your comfort and current back pain. In any event, I don't think it's necessary at all after breast augmentation to adhere to such rigid positioning restrictions that we cause other problems. Best of luck.
Contact your Plastic Surgeon regarding sleeping on your back. In the early healing phase, I have no problem with my patients sleeping on their side with pillows for support as long as they wear a tight support bra.
I think the best way to avoid pain from sleeping on your back is to avoid sleeping on your back. If your plastic surgeon was the one who gave you these instructions, then you should get back to them and ask if they think it is reasonable for you to change your sleeping habits.