This came through the ESRI higher-education list-server:

For the record, I posed the question to the Esri Geodatabase team. They confirmed that zipping a File GDB is what's needed - there is no tool inside ArcGIS to do it.

Caution: When you do ZIP a file GDB, you should always use ArcGIS (NOT Windows) to make a copy of the FGDB first, and Zip the copy. This will ensure that any locked files are released in the FGDB you distribute. (If you or your students get errors and the FGDB seems to be "corrupted," locked files are a likely culprit.)

There is not a problem with creating the .zip file and posting it on a FTP site. You probably won't see much in the way of compression with imagery. The download from the FTP site really depends on the connection speed.  A slow (DSL) connection will possibly timeout before the transfer in completed.  A faster (T1) connection should be fine, but it will still take awhile.  Just have to give it a try.