Can I use TimeMachine with a Thunderbolt hard drive? The Apple technicians told me that I needed a specific device to use Time Machine... namely a wireless device that had a relatively small hard drive.

Since I use Oracle Virtual Box and have very large files, I think it would be best to backup using Thunderbolt, instead of Wifi.

Can I use a normal Thunderbolt harddrive and have Time Machine work with that?
Yes. You can use Time Machine with any directly attached hard drive.  It is much faster, although if you have a laptop, it is somewhat less convenient to remember to plug the drive in than to use wireless.  

The folks from Apple were probably recommending a Time Capsule, which is a wireless router with a built-in drive for backups. It's definitely an easy way to do it, but not required.