I had the same problem, Googled for an answer, and came up dry. This thread--which lacked a resolution--was the closest I found. And since I now know the answer, I decided to respond to this very old post.
The short answer: If you have this problem,
you need to install the Flash plugin for Internet Explorer. You need to do that because Media Player Classic uses the Internet Explorer Flash plugin to render SWF files.
I use Opera primarily, and sometimes Firefox. Even when I had the problem with MPC not being able to play SWF files, Opera and Firefox could open Flash just fine. This is because the Flash software for Opera/Firefox is totally separate from that of Internet Explorer/Media Player Classic.
So, using Internet Explorer, go to http://www.macromedia.com and follow the links to install Flash. You do not also need to install Shockwave.After that, MPC will happily play SWF files.
I almost forgot to mention that after installing the Flash plugin for MSIE, you'll need to re-assign the SWF file association in MPC, if you had it set to begin with. (MPC > View > Options > Player > Formats > Shockwave Flash file)