From the point of view of the gravitational force I don't think there is any obstruction to having a black hole with color charge, but due to confinement it is not really possible to built up a macroscopic amount of color charge. An isolated quark will pull anti-quarks out of the vacuum in order to have an overall color neutral state.
The statement "from the point of view of the gravitational force there is no problem" is not trivial though. There are situations where you might naively think the black hole should have some kind of charge, but gravity prevents this from happening. In particular, it is widely believed that there can be no global symmetries in quantum gravity. If you built a black hole with a non-zero amount of baryon number, for example, the black hole would evaporate into a state with zero baryon number.