Get a router that is designed to be used outdoors and skip the waterproof box. You'll spend a little more money, but you will get better results. Higher end routers put off a decent amount of heat, especially if their transmit range is set high.
If you are set on the waterproof box, make sure there is some ventilation to allow heat to escape. In your house you probably don't notice the router getting warm, in a sealed box you would. Bugs will appreciate the warmth the router gives off, so make sure you keep the box clean of bugs. Some bugs really enjoy chewing on electrical wiring and components.
-Michael