Is it a cell's color that enables it to carry oxygen or not?

Yes: All cells can take in oxygen; a normal part of necessary cellular activity. Oxygen is carried by hemoglobin and myoglobin which are red in color. It's not the color that makes the difference but the fact that hemoglobin and myoglobin happen to be red in color. Oxygen carrying molecules in other species are of other colors. It is the molecular structure which latches onto the oxygen.