Specifically, a relay that has "normally closed" contacts. When the relay is energized, the contacts are opened (the circuit is broken).
Yes, that would make for a very simple 'black box'. It would have a 12vdc input jack from the wall wart transformer and a 12vdc output cable with a plug that fits the router's power input jack. Inside the box could be a any simple "bare bones" type Arduino board that will accept 12vdc power (that is, has it's own internal +5vdc regulator).
The Arduino would then drive a small SPDT +5vdc relay, such as ( http://www.allelectronics.com/make-a-store/item/RLY-496/5VDC-SPDT-REED-RELAY/1.html ) that can be driven directly from a Arduino output pin. Do wire a reverse connected diode directly across the relay coil terminals. The relay coil's two terminals will wire to your assigned output pin and the Arduino ground pin. Turning the output pin HIGH will energize the relay, causing the normally closed contact to open, thus removing power to the router until the output pin is brought LOW.
The +12vdc input wires to both the Arduino power input and to the normally closed relay contact. The power input jack ground wires to both the Arduino board and the negative wire of the 12vdc output cable/plug. The relay's common contact wires to the positive output wire of the 12vdc output cable/plug.
I would just use a simple sketch using the millis() function to initially turn on the relay coil for 30 sec, then off, and then again on every 24 hours worth of millis() time for 30 secs. You could wire an external reset switch to the box wired between ground and the Arduino reset pin, so you could force a immediate 30 sec 'power off' state and then start the 24 hour wait for the next automatic power interruption.
Good luck with your project, and giving Windows 7 a little Arduino helper 
Lefty