I understand that you want to know if the Slingbox will work if you connect it to a different router.
The answer is yes. Keep in mind that the router (any brand and model) are designed to provide the IP address any network device needs in order to reach the Internet service.
This means that when you are at a remote location, you only need to connect to the Internet with your computer or Mobile device in order to log in to your Sling Account. This procedure will allow you to get the remote connection to your Slingbox. The following link will provide you with further information in order to know how the Slingbox works.