Is it possible to get shin splints in only one of your legs?

When a person suffers from shin splints as a result of stress-related imbalances, it tends to affect one leg only, rarely both. People lead with their dominant leg when moving; if they're left-handed, they lead with their left leg. Running or other activities can induce an imbalance between the muscles in the front of the leg and the calf muscles, resulting in shin pain (in one leg more than the other).

Is such cases, Rest from activity and get one of those custom orthotics to support your foot and restore balance between the two legs.

It is certainly possible to have pain in one lower leg. "Shin splints" is a junk term. Pain in the lower leg can have multiple causes and may occur in both legs or just one depending on the cause.