The AirPort Express cannot "extend" the wireless signal provided by the Verizon wireless router if you are trying to do this using wireless only.
But, you should be able to configure the Express to "Join" the wireless network and enable the option to activate the Ethernet port on the Express. The Express will not provide additional wireless coverage in this type of setup.
Configuring the Express to "Join" the Verizon wireless network may be tricky because you must know the exact type of wireless security that the Verizon router is using in order to provide the correct settings to the Express.
