Verizon won't be any help supporting non-Verizon router setups. Their techs will not touch non-Verizon equipment. It doesn't surprise me that you've gotten conflicting information from them about non-Verizon routers. Ask other users (as you're doing here), or visit:
The latest FiOS ONE set-top-boxes give you options. The main box requires a coax connection for linear TV and MoCA LAN. The mini boxes are pure IP devices that connect to the main box over the LAN. The mini's can connect to the LAN via MoCA (coax), Ethernet or WiFi. A Verizon router provides all three options and the box figures which it's connected to when it powers up.
These boxes work best with a Verizon router. A Verizon router is required for activation. If you want to use a non-Verizon router and have FiOS ONE boxes, the easiest setup is to put the non-Verizon router behind the Verizon router. This will give you a setup that works with the set-tops and is fully supported by Verizon. Other setups are possible with varying degrees of complexity and self-support; consult the above link for more info.
Note that the WAN (or internet) connection is a different matter. If you currently have a speed of 100Mbps or lower, you may have a MoCA WAN coax connection between the ONT and router. This is a different connection than the MoCA LAN link used between the router and set-tops, but shares the same coax. If you're upgrading speed as well, you'll need an Ethernet cable between the ONT and router, as MoCA WAN is not fast enough for speeds above 100Mbps. The router still needs to connect to coax to create the MoCA LAN for the set-top(s).