OzTm Only if the router's input voltage is 5V (check the original adapter's output voltage) and the powerbank can deliver enough Amps (usually around 1A should be enough to power it up, but that's just a guess; it really depends on the router's specs). In theory, the powerbank should have clear specifications, and it should also have overload protection, but even so you should check for overheating after a few minutes when you try it.

OzTm Yes, I just installed it today and I can confirm that the thread size is 3|8

OzTm Being "for Arduino", I would assume that these modules are made to work when supplied with 5V, not 3.3V. The red LEDs usually require a little less voltage than other colours to light up, so that's why yours is working.