USB to Ethernet

I see there are many USB to Ethernet Adapters available, but all of them are advertised to connect your laptop or desktop to a network. Will they work to also connect an external drive to a router via the drive's USB to a router to make a network drive?

No I don't think that will work at all. You need a NAS enclosure.

Yeah, USB network cards are network adapters just like PCI cards, your drive would need to be running some firmware that made it able to be networked for this to work (probably impossible) but you can find network hard drive enclosures pretty cheap, just get the one the same type as your drive (PATA\SATA) and stick the drive in from your existing enclosure.