Yes, you can customize mouse actions.
How to do so would depend on your desktop environment and the type of mouse. As far launching the screensaver, I'm not clear whether you mean launching the screensaver configuration dialog or the screensaver itself. If the latter, my guess--and it's only a guess--is that would need to somehow trigger the screensaver "Preview" if you want the screensaver to take effect immediately. The man page for whichever screensaver you are using should tell you more. (I always use xscreensaver.)
A web search for
customize mouse actions linux will turn up a number of articles and tutorials.