No, you won't have problems. You can connect any router you like behind a Verizon router. If you connect Verizon LAN to Nighthawk WAN, you'll create another network within your home, which works for many people. If you'd rather have one network, connect LAN to LAN and put the Nighthawk into bridge mode. Consult it's documentation if you don't know how to do that.
You mentioned you had gaming issues when using the Verizon router. If you care to describe these issues, I and others here might be able to help resolve them. This could eliminate the need for a 2nd router.
Start a new topic or ask questions in the open forum.
