Never heard of a printer automatically turning itself on just for printing! Usually they (HPs included) have a power-on button somewhere, which needs to be turned on before printing.
Are you sure they didn't have it switched on all the while with the previous PC? Can you reconnect to the old PC to confirm whether this is the case or not?
If all fails, can they not just power it on before printing? That's what I have to do with my HP Deskjet 640c. G