Why should Zuckerberg unnecessarily antagonize one half of the voters by becoming a party's candidate? No, Zuckerberg's best bet would be to run as an independent. Both parties are in shambles anyway and would be more a liability than a help for someone like him.
Zuckerberg is too smart to think a third party candidate has any chance of winning. The parties being in shambles does nothing to change that, and if anything it's going to heighten tribal affiliation with those two labels.