I have actually heard of a wireless router causing the printer to become unresponsive and I have heard of surge protectors causing printers to display false error messages. I do have a few more questions in case I need to help you in other areas as well.
1. What is the operating system of the main computer connected to this printer?
2. What is the printer's IP address?
a) Ethernet or Wireless?
3. What is the name and model number of your router? Sometimes even the router needs a firmware update.
4. Did this happen after you did the firmware update for the printer or have you done the firmware update yet (dependant upon operating system usually)?
5. Is this printer plugged into a surge protector or a wall outlet?
6. The part that interests me the most that I need clarification with; when the printer causes the router to shut off, does power-cycling the printer fix this or is that incident intermittent?
Answer those questions for me please so I can determine exactly where the problem is. With all of the many problems it is still hard to determine if this is a hardware problem, issue with firmware, or a network problem.