Need more info.. all SWF's from all sites?
If you right-click where the 'checkerbox' is what does the menu show you?

from my tests so far, it is only on this one particular terminal server and it
is with all SWF's from all sites. When I right click on the SWF I get the
flash player settings with {Zoom In, Show All, Quality, Settings, Print, and
About Adobe Flash Player 9...}

Hmmm.

The fact that you can see the entire Flash Player context menu is good. That
means the SWF content is entirely loading (I expected to see 'movie not
loaded'...)

If it's only one machine, I'd have to conclude it's some kind of video driver
problem. Check to see if there are any newer drivers for that video card (or
try different settings for the existing driver).

The problem is not on only one machine. It is on the terminal server and
several other computers. It looks like I am also having problems with gifs.
Some of the sites that I have checked are:
http://www.playproject.org
http://www.regionalautismcenter.org/