Yes, actually, they can.
The rules for when an employee can come on and off an employer based medical plan are set by Federal law. When you are a new hire, either you come on the plan the first day you are eligible or you do not come on until open enrollment. Period. No exceptions. That is determined by IRS regulations, not by your employer. It is not even remotely unusual for a new employee's first month of coverage to be retroactive.
If you paid out of pocket for medical care that would have been covered by the insurance, then apply for reimbursement from the insurance carrier. Trust me, you won't be the first to do that. You probably won't even be the first person today to do that.
Two things I am tired of typing: 1.) A wrongful termination does not mean that you were fired for something you didn't do; it means that you were fired for a reason prohibited by law. 2.) The above answer, whatever it is, assumes that no legally binding contract or CBA expressly says otherwise. If it does, the terms of the contract apply.