WinXP is the only M$ OS I know of that includes its own unzip utility. Win95/98/Me never had a default unzipper. Most people used WinZip as the earlier versions had a free trial with no time limit. Unfortunately this is no longer the case. The older versions (up to WinZip 8 I think) will still work if you already downloaded one.
PS. Your PC might have been supplied with WinZip already installed. You might find an earlier free version on a magazine cover disk.
I don't know what it is or where it came from, but it's been on the old PC from new (it's still there - I still have the old PC set up). It's called the "Unzip Wizard" and when you right click on the folder, you get this:

I've never installed any type of extractor because, with this, I never needed one.
I've seen the 'Winzip' program (on PC's at work). I think that has a different icon for the folders (I think it's supposed to be a picture of a 'vice' of some sort). The 'stock' program I have on the old PC just uses the 'yellow zipper' type folder.
So this thing definitely isn't Winzip and it came new with Windows Me, on the OEM disk from Compaq.
Anyway, regardless, what you're saying is that these zip files aren't being recognised because the Me OS doesn't come with a default unzip program?
Thanks