Add a Flash file as a movie:
In Flash export the file as a video file (.avi will work)
Upload the .avi file to your Google drive
While editing your page in Google Sites Insert>Drive> click on your movie
Be patient, Google has some work to do to the file
OR add a .swf:
Upload your .swf file to your Google site.
More
Manage Site
Attachments | Upload
You will need to note the file location
Create your new page
Select <HTML> from the tool bar
Copy and paste the following code on to the HTML code window
<embed xmlns="http://www.w3.org/1999/xhtml" pluginspage="http://www.macromedia.com/go/getflashplayer" src="http://sites.google.com/site/shdbtt1o/google-sites/skySun.swf" width="540" height="600" />
you will need to edit the code to match the names of your site and file
you may need to change the height and widths also
The above code is used on this page
Note: This solution was orginally found at FlashPerfection