Binary hydrides formed by the pnictogens of phosphorus, arsenic and antimony are studied at high pressures using first principles methods. Stable structures are predicted and their electronic, vibrational and superconducting properties are investigated. We predict that SbH$\mathrm{<mrow\; data-mjx-texclass="ORD">4</mrow>}$ and AsH$\mathrm{<mrow\; data-mjx-texclass="ORD">8</mrow>}$ will be high-temperature superconductors at megabar pressures, with critical temperatures in excess of 100 K. The highly symmetric hexagonal SbH$\mathrm{<mrow\; data-mjx-texclass="ORD">4</mrow>}$ phase is predicted to be stabilized above about 150 GPa, which is readily achievable in diamond anvil cell experiments. We find that all phosphorus hydrides are metastable with respect to decomposition into the elements within the pressure range studied. Trends based on our results and literature data reveal a connection between the high-pressure behaviors and ambient-pressure chemical quantities which provides insight into understanding which elements may form hydrogen-rich high-temperature superconducting phases at high pressures.