I've only ever heard of something similar once and in that case the router didn't like the uPnP feature of the new patch.
Your issue sounds a bit different but maybe it's worth a try disabling it anyway and see what happens.
You do need to forward your ports manually then of course.
To disable the uPnP for the patch find the file "oledlg.ini" in your WinMX folder and open it with Notepad.
Find the line that says /enableupnp 1 and change that to /enableupnp 0.
Then run things a bit and see fi that fixes things. If nothing else it helps excluding it
Also when you say you can't get to the internet from that computer could it be that your dns isn't working then?
That would explain why you can access your router fine as you're using the IP addess for that.
Maybe the next time it happens you can try to access http://18.104.22.168/
and see if you get there, it should be the winmxworld home page on the backup server
PLease report back on what happens...