The Ethernet ports on the Verizon set-top boxes cannot be used to connect devices. With the exception of the FiOS ONE mini boxes, they are disabled. The FiOS ONE mini boxes can use them to connect to the LAN like any other wired device. The do not function when the box is connected to the LAN using MoCA or WiFi.
If you want to get an Ethernet connection to your smart TV but don't want to run a cable, consider using a MoCA adapter. Use a splitter to provide coax to the set-top-box and the adapter, and you'll be good to go. Verizon sells a suitable model for $55 here: