What would cause severe pain if pressure is applied just under the lower rib on my right side on my back?

Chest wall pain: Could be as simple as costochondritis verses more complex problems as pleurisy or shingles.

Kidney: The area just beneath the back bottom rib and near the spine is called the "costovertebral angle", and if there's an issue with kidney on that side (infection, stone, etc.) that is where you would feel the pain. If it persists, please get checked by a doctor.