So this is the second time Verizon has dinged me for uploading my own music to Google Music which is supposedly legal. I think my ripping software must have named the files based on some algorithm Verizon doesn't like because I've only uploaded two albums in the last few months and both were reported as file sharing copyrighted material. Other than changing how my files are named is there any way to stop this madness?
Ripping music from CD's that you own and uploading them is generally considered legal as long as it is for your own use and do not share the files.
Use HTTPS and encrypt your stuff with a VPN if you have to. Anything that involves "monitoring" the Internet I've found has large problems detecting valid traffic. It's not Verizon's job to police the Internet.
Is the Verizon notice you're receiving this copyright notice about detailing who is sending the letter?
I would also do a double check of your network to see if anyone is logging into it that shouldn't be, or if anyone is downloading stuff using "Free MP3" apps or programs on smartphones, tablets, or computers.
I uploaded 8,000 tracks to Google Music a few months ago with no issues at all (other than Google Music replacing a few Explicit tracks to "safer" versions, which is fixable within Google Music.) My track names couldn't have been more obvious. So I'm not sure what your issue is, unless it's something very new at Verizon.