A compromised router (which is just another computer that happens to be on the network path between your other computers and the Internet) can attack hosts and compromise them. It can also redirect your HTTP (not HTTPS) requests to malicious websites in an attempt to fool you into installing malware. However, I would be really surprised that it could compromise a brand new, up to date Windows machine assuming the user wasn't an idiot and didn't intentionally execute the crap served from the malicious websites it redirected to. I'm not saying it's not possible - there is always a risk of a zero-day vulnerability (which would work silently without any user intervention), but nobody is going to "waste" a valuable zero day on you unless you're a really valuable target (and then they'll do a better job at silently compromising you rather than alerting you because your Internet doesn't work).
I'm actually sure your problems aren't caused by malware, but given that both PCs seem to have malware (at least according to the so-called "technicians") it looks like both computers continuously compromise each other so it's time for a proper solution.
Stop paying incompetent people for installing useless "best virus protection" (there is no such thing, antivirus evasion is trivial and the best antivirus is you) on an already compromised machine. Follow the usual procedure for dealing with compromised hosts : from a known clean machine (friend's PC, etc), download the ISO corresponding to the Windows version on your computer - pirate sites are a good source for them but make sure to verify the hashes against the official hashes published on MSDN to make sure you didn't get a compromised version. Now burn that on a disc, bring it home, disconnect your computers from any network (Ethernet disconnected and Wi-Fi switch set to "off"), boot from the disc, nuke everything and reinstall. Do the same for all machines on your network, then reset your router, connect the machines to it and only let them install updates - don't browse the Internet just in case your router redirects to malicious websites that would exploit the still unpatched Windows. Once it's up to date, if you still see weird stuff like redirections to malicious pages (sometimes full of nasty ads like porn), return the compromised router to your ISP and invest in a better one that has at least a somewhat decent firmware.
Note that you should never trust the "Refresh My PC" feature nor the built-in recovery partition for reinstalling a compromised machine - writing to that partition is trivial and malware can easily use it to persist.