Can you see the EULA from this link? And does it show the sort of aetherflow/starry background graphic and colored text? Those are basic HTML elements and should render just fine provided there isn't a security/connectivity issue blocking access to the flat files and such.

http://frontier.ffxiv.com/arr_win/eula.html

Also, if you hover over the next button does it change it's shade slightly and also "light up" if you click the box for accepting the agreement? If you actually click the "Next" button it will likely return strange errors because you don't have all the stored values for the scripts, but it should at least be able to run those mouse-over and on-click portions of the page's scripting. If not... there is something amiss in the browser--possibly the security settings need to be lowered or have the site added to the trusted sites list.

And yes, the launcher is based loosely on MSIE--it doesn't use your default browser, but it's own sort of custom version of MSIE, but it calls upon the supported files that would be installed/registered in the Windows System folders and such for MSIE and various browser technologies (like javascript or ASP and such). In essence, something being out of whack within MSIE's config/support could impact the launcher if it is something functionally bound at the system-wide level (like .NET and Java type stuff).

On that note... just something to try to see if it may have an impact. Go into your Java properties from Control Panel and go to the Advanced tab. A few categories down the list there should be an entry for which browsers that Java will run on. It should have Internet Explorer checked, possibly even grayed out as well. Make sure there is also a check in the box for Mozilla as well. MSIE was basically built around that initially (as well as a long list of other browsers like Firefox and such). Enabling it is a workaround for many alternative browsers to MSIE. Not too far a stretch for the launcher to be affected by it as well...it may simply be something is breaking the scripting, and that may get it parsing things again.