The Service Set IDentifier (SSID) only affects your wireless clients ability to find the WiFi service.
Once you have the local viewing setup for your Slingbox your done with its configuration. The only thing that's preventing you from remote viewing is your router.
Slingbox provides its support through port 5001 (by default) and that's how you access it from your private network and public network. However, your router has certain services to provide you with a safe networking enviroment. Unfortunately, these services are also preventing you from accessing your Slingbox remotely. One way to allow your Slingbox outside of your network is port forwarding. Without port forwarding any internet request for port 5001 will be ignored. This is because your router doesn't know which of your networked clients will handle it.
When you configure port forwarding on your router, you specify port 5001. This is the port which the Slingbox provides its services. So, the next important part of completing the port forwarding rule is the IP address of your Slingbox. Once you've completed setting up the port forwarding rule, all of the outside requests for port 5001 to be directed to your Slingbox.
Verify the information of your port forwarding rule. Using the stand alone Slingplayer, go to the command bar | Help | About Slingbox <model> (Alt + E) | Information Tab. There in the "Lan" subsection, you'll see the "IP Address" and "Port" information. That is the information that you need to setup Port Forwarding.
Another alternative is Universal Plug and Play (uPnP) its a set of protocols that allows to devices to communicate with one another for configuration purposes. If your router is uPnP compatible, it is possible that its services is disabled by default. If you can find the uPnP configuration within your router's configuration menus then enable it and run through the Setup Assistant for Remote Viewing.
To test your router's configuration, go to http://www.canyouseeme.org and in the "What Port?" box type in the port number that your slingbox is using and click the "Check" button.