The photon emissivity from the bremsstrahlung process ee-> eegamma occuring in the electrosphere at the bare surface of a strange quark star is calculated. For surface temperatures T<10^9K, the photon flux exceeds that of e+e- pairs that are produced via the Schwinger mechanism in the presence of a strong electric field that binds electrons to the surface of the quark star. The average energy of photons emitted from the bremsstrahlung process can be 0.5 MeV or more, which is larger than that in e+e- pair annihilation. The observation of this distinctive photon spectrum would constitute an unmistakable signature of a strange quark star and shed light on color superconductivity at stellar densities.