Using Firefox in Eclipse instead of IE
The SWT Browser widget that is uesd by the internal web browser chooses the platform default for its web rendering technology. So for Windows that is the MSHTML COM component (IE renderer). For Linux that is Mozilla (Webkit for 3.7 is available) and on Mac, WebKit(Safari). However, there are third party plugins like MyEclipse that do embed a mozilla runtime (XULRunner) that do use the Mozilla (Gecko) renderer for a browser.
So to answer your question there is no way to use Firefox as internal web browser on windows, however, you can install 3rd-party plugins for Eclipse (MyEclipse, JBoss tools, etc) that embed Xulrunner binaries that will allow for embedding mozilla renderer as the technology used by an internal Eclipse browser. Note, this may require that use the 3rd-party web browser widget instead of the standard eclipse platform one, but those 3rd party tools usually provide a toolbar action, etc.
Install Ajax Tools Framework plugin. It will automatically come with a recent version of XULRunner (Mozilla/Firefox engine) and will set your default internal web browser to it.