Hello and welcome to JustAnswer.
I'm sorry to hear about your situation and hope I can help.
Unfortunately, until you are done with your probationary period, you are an at-will employee of the county, meaning that you can be terminated at any time for any reason (or none at all) with or without any prior notice.
That said, although it appears your termination was not justified it would not give rise to any cause of action against the county. The only factor that would change that is if you were terminated due to discrimination based on a protected characteristic, such as age discrimination in violation of the Age Discrimination in Employment Act.
If you have any reason to believe that is the case, you should file a charge against your employer through the Equal Employment Opportunity Commission, which you can do following the instructions online here:
The EEOC will investigate the complaint on your behalf and either file charges on your behalf or issue you a right to sue letter so you can pursue your claim against your employer through a private attorney.