All the information for an EagleFiler library is stored in the library folder that you created when you used the New Library… command. (See also the Where is my EagleFiler library? section.) Simply make sure that your backup software is set to include your library folder. There are also various ways that you can sync a library folder to the cloud.
Note: If you are using an encrypted library and want to back it up in encrypted form, you should instead back up the .sparsebundle or .sparseimage file and make sure that the library is locked/ejected when you do the backup.
Additionally, you should back up EagleFiler’s support files in the folder:
/Users/<username>/Library/Application Support/EagleFiler/
and its preferences file:
/Users/<username>/Library/Preferences/com.c-command.EagleFiler.plist
To access these items, see How can I open the Library folder?.