The standard Flash Player is a browser add-on, and not a standalone player. So what actually opens when you open an SWF file at home? Flash Player Projector?
If so, you also need to download it at your school, from http://www.adobe.com/support/flashplayer/downloads.html, then do the file association.
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Yes, that is the Projector. Download it from the link I posted earlier, then do the same as you described in your original post (right-click and "Open with...").