What files do I need to backup if I have to re-install?

You should back up your "work folder" as it contains your performance history. Although the s/w will work fine even if you don't back it up.
See this image to see the location of the work folder:
You can simply zip up this folder to back up your test history. Unzit it and set the unzipped folder as your work folder through ETS Viewer, if you ever need to see it again.
You should also backup up etsviewer.jar and the question bank file (the one with extension .ets). Both can be downloaded from our site as well.
After installation, it will ask for the license. You should also backup the license from the current installation by going to Tools -> Question Bank License Information (Manual). Just copy and paste the license to a text file and use it later.