We present a theory for the nonequilibrium current in a mesoscopic Josephson junction which is coupled to a normal electron reservoir, and apply it to a chaotic junction. Large sample-to-sample fluctuations of the critical current Ic are found, with rms Ic�sqrt[N]e�/ħ, when the voltage difference eV between the electron reservoir and the junction exceeds the superconducting gap � and the number of modes N connecting the junction to the superconducting electrodes is large.