I'm mostly posting this for my future reference, as this is the third time I've had to fix this issue. The basics are this:
Since FireFox 3.6, Mozilla added a plugin container that runs plugins in their own process. This prevents nasty browser crashes when the plugin crashes. Overall, it was a good move on Mozilla's part.
However, for Flash and Flex developers, this can represent an issue when debugging your movie. When in debug mode and you have set a breakpoint, FireFox will think the plugin is hung if you have it paused for more than 15 seconds. When this happens, FireFox will kill the plugin container process, crashing your debug session.
The fix to this is easy. Go to about:config in your FireFox address bar. After confirming the disclaimer, you can set the dom.ipc.plugins.timeoutSecs to the value you want. A value of -1 will turn this off completely and will not timeout the plugin. The downside to this, of course, is that it will not kill plugins that are actually hung. It is the main reason I now use Chrome for my primary browser and FireFox mainly for testing.
I found this fix at http://support.mozilla.com/pa-IN/questions/705281