Is my A/C circuit break worn out?
If the breaker is sized accordingly for the system running on it, then no, it should not be tripping. That said, yes, breakers can wear out. How old is the breaker?

Next question, Does the wiring for the A/C near the box feel warm? It really should not. If it does, I suspect a loose connection or HIGH resistance connection between the A/C and the breaker. It is possible the wiring in the panel is loose, but the HVAC company should have checked that.

Did the old unit trip as well? it is a sign something is wrong with the wiring or the breaker, and if you told the HVAC company that, then they should have checked or one of you called an electrician before connecting the circuit.