For part of yesterday my previous router had assigned public IP addresses to hosts on the LAN, as if it were passing through DHCP service from Verizon’s edge router. I know I don’t want my hosts exposed that way, so I changed the router. Could my systems have been hacked in that time? Would clamav tell me? Am I in the right forum?
When your systems could be connected direct from the internet without any firewall, the were vulnerable to all hazards from the internet. Of course local protection on the systems themselves (like not listen on unneeded ports, Firewall) are very important in such a case.
I do not see why you mention clamav. As far as I know clamav tries to detect viruses for MS Windows systems. You running Linux are not endangered.