Note that this is a peer-to-peer forum. Verzion does not provide direct support here. If you want an official Verizon answer, you'll have to contact them directly (see "contact us" at the bottom of these web pages).
The box is called on ONT, which stands for optical network terminal. The battery is a sealed lead acid battery. It is really easy to replace. Just pull it out and swap connectors.
The primary safety message is don't open the battery as there is nasty stuff inside. Unless you plan to take a hammer, saw or prying tool to see what's inside, you're safe. You can easily recycle it at Home Depot or Lowes.
Note that if you do not have Verizon FiOS phone, internet or TV service, you don't need the ONT to be powered. You can remove the battery and unplug the ONT without concern.
Also, don't be surprised if Verizon sends you a bill for the battery. The battery is not Verizon equipment, it is owned by the occupant. If you truly don't have an account with Verizon, you may never see the battery they promised you. But no worries, just yank out the old one and unplug the ONT.
Enjoy!