Is there a breaker that trips when the power goes out?

Circuit breakers are not designed to protect from voltage surges. They only trip quickly for very high currents. I explained the time it takes for a circuit breaker to trip in this post.

We are assuming that the technician provided the OP with a correct explanation. A surge might take out the control board of an A/C unit, but that is repairable (although expensive - for an old enough system it might make sense to replace rather than repair). Damage requiring replacing the whole system sounds like a motor burned out due to a brownout (period of voltage too low for a motor to run but high enough to cause the stalled motor to burn out) rather than a surge.