We calculate the critical supercurrent through a resonant scatterer between disordered BCS superconductors with Cooper pair breaking. Profound deviations from the case of ideal BCS superconductors are found in line with recent experiments on the Josephson effect in carbon nanotube quantum dots. The effect of pair breaking is accounted for within scattering theory via the relation between the Andreev scattering matrix and the quasiclassical Green functions of the superconductors in the Usadel limit.