Greasemonkey, by itself, cannot automatically save zip-files, or anything else, to the local file system.
This is by design; allowing user/page JavaScript to save files is a proven security disaster.
Your options:
- Have Greasemonkey select the right link and open the File-Save dialog (saving you the search effort and 1 click).
- Have GM relay the zip file to your own server. Your server application can then automatically save the file.
Note that the "server" could be your own machine running something like XAMPP.
- Write your own Firefox Add-on.
Option 1, GM only:
What GM can do is pop open the File-Save dialog for the correct file:

User interaction will still be required, if only one click.
For example, suppose the page contains this link:
Click me, sucka!
Then this code will open the File-Save dialog for it:
var clickEvent = document.createEvent ('MouseEvents');
var firstZipFile = document.querySelector ("a[href*='.zip']");
clickEvent.initEvent ('click', true, true);
firstZipFile.dispatchEvent (clickEvent);
Option 2, GM and your own server application:
Greasemonkey can use GM_xmlhttpRequest() to send files to your web application -- which you will have to write. The web app can then save the file to the server automatically. You can set up your local machine to be the server.
For more help on this approach, read this and then ask a new question.
Option 3, write your own FF extension (add-on):
If you decide to take the Firefox add-on route, see "MDN: Downloading files".
For more help on this approach, read this and then ask a new question.