Shin splints is a term used to describe shin pain, it is not a stand-alone affliction. There could be several different reasons you are experiencing shin pain, the term shin splint (or tibial stress synrdome) is used to describe any of the conditions.
Swollen muscles, stress fractures, and flat feet can all be attributed to shin splints.
When the posterior tibial tendon gets stretched or torn the arch of the foot collapses. As the arch collapses the foot will over-pronate or turn outwards. This outward malalignment of the foot will stretch the muscles and tendons in the lower leg. This can cause shin pain.
So when looking at a treatment for shin splints it is important to understand what is causing the shin pain. A calf compression sleeve that provides warmth and a gentle squeeze may be all that is needed. But if the shin splint is being caused by flat foot then an ankle brace and foot orthotic might be a better solution.
When you speak with your physician make sure find out exactly what is causing your shin pain.