Time Machine won't use external drive

Not really. Could be the drive.

Are they directly connected to the Mac? I doubt a hub would cause any issues.

I don't know why it would just stop backing up to it. It would make more sense if it kept failing the backups and starting over. That would tend to indicate a problem communicating with the drive.

I'd remove it from TM, erase it, check it with First Aid, then add it back to TM if all checks out.