yes sir you can.
If you have the EEPROM.bin file, you canopen it on your PC using a prgram called Live Info. This displays the codes etc your EEPROM contains, such as your Mac address, online key, and your HDD locking key.
What I recommend you do, is install the xbox app Configmagic. Then make use of this apps back up feature, it creates an 'EEPROMbackup.bin', 'EEPROMbackup.cfg' and a 'EEPROMbackup.txt' file in the root of your E:. Then back them up to your PC, the txt file allows you to view these codes/info simply with Notepad etc.
If your dog did actually pee on your box totally stuffing it, you could copy the HDD locking key for the drive from the txt file, put the HDD in another box, then update the new boxes EEPROM (using the Live Info) with the HDDs locking key. One harddrive salvaged and a smack on the ar$e for the dog!
also, when you back up your EEPROM with the Xenium, it backs it up to the Xeniums own memory. The other backup option as you said is to back up to .bin where the EEPROM.bin file is created in the root of the E:.