What causes upper right rib cage/back hurt while breathing?

Could be: pneumonia, pleurisy, pulmonary embolism, pneumothorax, gallbladder attack, kidney infection, or even shingles. Impossible to tell for sure without exam and tests. Get to a doctor asap.

Many things: things in differential: costochondritis, rib fracture, pulmonary embolus, thoracic fracture, disc problem among other problems. Would need a better history to narrow down the possibilities.