What causes upper right shoulder pain that hurts in both the front and the back?

Arthritis: You probably have "shoulder rotator cuff" problem. You will possibly need an MRI of your shoulder and on orthopedic evaluation.

Could be many things: Shoulder pain can come from many sources. The joint due to arthritis or inflammation. The muscles to due fatigue or strain. The rotator cuff due to inflammation or a tear. The bursa due to hypertrophy or inflammation. Finally the nerves to the shoulder can be compressed in the neck and cause pain. See a board certified orthopaedic surgeon for evaluation and treatment.