Can my competitor fire his part time employees if they accept a part time job at my company?

Assuming that the employees have no contract to the contrary, and that the policy applies across the board, rather than discriminating against a statutorily protected class (e.g. women can't work for you but men can), I believe the answer is yes.  Although California's employment laws are generally very employee friendly, I am aware of no law which would require an employer to continue to employ an employee who works for a competitor.   I don't think being a "right to work" state has anything to do with this issue, as it  refers to whether an employee can be compelled to join a union as a condition of employment.