There are a lot of answers, which I can't narrow down without more info, but here they are.
If each PC has its own Zip drive, then the Iomega tools will be easiest for the users.
If the users' files are on the server, then it will be easier (both as implementation and in the long run) to do their backups for them. In such a case, the Iomega Backup utility is still the best bet - it can be automated, given compression options, and so forth. The only problem with this approach is that it's unlikely that 160 users' data will compress down to 250 MB.
PKZip is a wonderful program, but it's shareware and, as a government entity, you have to be very careful about using it that way unless you've paid for it. MS Backup and Iomega tools have compression options that work as well.
Have you considered the cost associated with Zip? Hard disks are very cheap these days -- the cheapest backup you could want is an extra hard disk with compression installed. You can make a desktop shortcut for everyone that leads to that person's folder on the Backup drive, and they can drop their data there. Or, you can write the batch file:
COPY "\My Documents\*.*" \\SERVER\BACKUP\USERNAME\ /y
and put its shortcut on the desktop. If that compressed drive fills up, announce that anything older than 1 year (or whatever) will be destroyed in 1 week (or however long you want to give them to salvage anything worth keeping).
Good fortune,
Don
