Most developers are not in position where the client would have slightest idea when developer came in.

If you have regular hours long outages that require specific developer to come in and fix the code asap, you have huge mismanagement problem. The developers coming not in the morning is your least problem. Insufficient testing, apparently badly set up servers+application and most likely general mismanagement are primary problem.

Also, if you are in this situation, then it is reasonable to assume that your developers had to deal with similar outages in the middle of the night or during weekends.