My Verizon MI424WR router resets and drops all client connections without my intervention

[#1]

There are several things that can cause random router reboots.

1. Overheating

2. Bad power supply or bad outlet/surge protector. 

3. Some other physical defect in the router.

4. Least likely is a setting in the router like QoS.

Check for physical issues and then replace if unresolved

[#2]

I was having this same problem.   On a brand-new FIOS installation, the brand-new router would reboot itself many times an hour.  Verizon replaced the router with another one and the problem persisted.

My setup includes an Apple Airport Extreme that is running in bridge mode from the MI424WR.

A tech came over and found a cable was connected from a cat-5 output port on the Airport back to one of the output ports of the MI424WR -- a "loopback."    He said he's seen this quite a few times and it's now the first thing he checks for.     This had apparently been done during the initial installation and switchover from Cablevision, when cables were being disconnected and reconnected.

Things are running fine now, and I might wind up removing the Airport from the setup.

[#3]

Good old network loops. I'm surprised at just how little consumer gear supports Spanning Tree Protocol. It's not just for big networks!