Is there a legal recourse ? Can I file a wrongful termination complaint ? Do I have to do arbitration, particularly since he did not sign the form, there is a space for his signature ?
A: Confidentiality agreements are enforceable under California law. That should not pose much of a problem for you, because there isn't much that you could reveal other than the customer lists that would injure your former employer. So, you don't contact any customers, who don't contact you first.
Covenants not to compete are absolutely void under California law (Bus. & Prof. Code 16600). So, you can tell your former employer to "stick it where the sun don't shine" (for lack of a more colorful phrase), because no court and no arbitrator will enforce the covenant against you. You can work where and when you wish, and you cannot be forced to pay liquidated damages in any amount -- period, end-o-story.
Re wrongful termination, Labor Code 2922 permits an employer to terminate an employee "at will," unless there is a written contract with a specified termination date of at least one month in the future. Your contract was expressly "at will," so, generally speaking, you have no grounds for a wrongful termination action, because your termination wasn't wrongful -- i.e., the employer didn't need good cause to terminate you.
Since you weren't terminated for misconduct, you can file for unemployment benefits, despite your being a part time employee.
That about covers all the issues. Now that you know that your former employer can't stop you from working, your future should feel a bit brighter.
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