I have a good idea of whats up here. Just to re-confirm, your nighthawk is connected, and you can access the internet from it?
I believe the problem is that your Nighthawk is handing out LAN IPs in the range of 192.168.1.X, which is also what the G1100 hands out, so the G1100 and Nighthawk network are overlapping.
You'll need to change the IP range that the G1100 uses for the set top boxes. To do this, please login to the G1100 (Easiest way to do this would be to plug an ethernet cable from the G1100 into your computer). Go to http://192.168.1.1/ and enter the username and password listed on the side of the router.
Once you've logged in follow these steps:
Click My Network at the top of the screen. Then click Network Connections on the sidebar of the page that will appear.
Click on Network (Home/Office).
Click the Settings button at the bottom of the page.
Look for the portion of the settings page which says Use the following IP address:, you should see 192.168.1.1. This is what you have to change.
Change the address to 192.168.5.1.
You just need to change two more settings before moving on. Scroll down a bit to the IP Address Distribution section. You'll see a "Start IP address" and "End IP address" section. Change these from 192.168.1.2 and 192.168.1.254 to 192.168.5.2 and 192.168.5.254.
Your configuration should look like this (images may not always show up due to them needing moderated manually):
Lastly, click Apply at the bottom of the page. The Verizon router will reboot. Reboot your set top boxes, and check for connectivity.
I'm 99% sure this should solve your issue