When I open a real-time viewer, I see a toolbar with zooming options and a big green box to the far left. What is the purpose of the red box?

Red is bad. When you see the red box, it means that the real-time viewer could not subscribe to all tags that are configured on the HMI screen.
This typically happens when an HMI screen is moved from one project to another, and now the new screen is trying to subscribe to the tags that are configured in the screen but the tag names are different in the new server.
This could be caused by errors or misspelling or removal of tags from the database. You can see the list of missing tags by moving your mouse of the red box or you can click on Ctrl F3 to see a subscription list for the active window. The subscription window shows all subscribed points in green and all failed subscription points in red. A good configuration should not have any red line in the Ctrl F3 window or show the red box.