I'm having a hard time visualizing exactly what would happen. What similarities does it share with our usual superconductivity and what would be different?
Ordinary (electronic) superconductivity is characterized by 1) the Meissner efffect (screening of magnetic fields), 2) a gap in the excitation spectrum of single electrons, and 3) perfect conductivity (zero resistance),
Color superconductivity in dense quark matter is characterized by 1) the Meisner effect (screening of color magnetic fields), 2) a gap in the spectrum of non-Goldstone modes, and 3) (depending on the exact nature of the color superconducting order parameter) superfluidity (flow without viscosity). Note that there is no gauge invariant color charge, so it is not clear what one might mean by perfect color conductivity.