It's not flash, and it's an old issue to which there's no simple answer. Basically, if you need to render an html page, you need an engine to do it. This is different from simply showing a link(which can be associated with anything), it needs to actually render *in* the MOTD popup, not in a browser. The only engine that ships 'for free' on Windows is, you guessed it, IE. Firefox doesn't ship with a callable render engine which can render in custom windows, and even if it did, you'd now be stuck following their API, which would likely differ from browser to browser.
The only answer is to bundle a free browser, like gecko, in with the application and double the download size just in order to render an MOTD. That's not likely to happen.
Other apps do this too, if they don't, they're a large app with a bundled html renderer.
I would consider following the newsbin twitter, it's linked in the upper right of this page. It works much better for getting more frequent news, plus release announcements. Of course, the messages are shorter.
DT