Powering up and down of a microcontroller doesn't damage it. It is important to take the rest of the circuit into account, as it is possible for things that the microcontroller is connected to cause issue to themselves or to the microcontroller. This is not normally an issue if the hardware engineer has put in the appropriate fail-safes for when e.g. the microcontroller pins are at undefined states.
Your new software guy should provide a reason for his opinion, otherwise it's as good as baseless superstition.