I had a tech come out to check my lines. I explained to him very carefully the extent of the problem, that I had been recording times for two months, that this was the third Rev I, and that the Rev I router was certainly rebooting on it's own. He made a call into home base and came back with a Rev F router which has been running for 4 days now. I suspect this will resolve my issues.
In addition, he replaced the power supply and battery backup on the ONT. He said that verizon was replacing the old style as they encountered them.
I recorded log files from my router using a free webapp called http://papertrailapp.com/ No one from Verizon wanted to look at the log files so it might have been overkill.