I did a Google search for embedding flash files into Google sites and came up with the instructions at How to embed Flash SWF file to Google Sites?. Google has changed a bit since this was written but basically it still stands. Upload your SWF file into your Google site's file cabinet and right-click on the "view" option to get the URL. Put this URL into the <embed> tags as outlined in the linked page.
In case the link breaks the <embed> tag looks something like this:
<embed src="http://sites.google.com/<YOURSITE AND SUBFOLDERS>/<FILENAME>.swf" width="XXX" height="YYY"></embed>
Edit your page and select the HTML editing option to insert this.
This solution helped me as it means that I didn't have to go down the root of making my stuff public as it is for internal use only.
Hope this helps,
Greig