Marc,
Sorry didnt realise that the clear button was not in the MGRS tool, new .swf file with it included.
I'm not sure what is happening with the coordinate tool not showing the right grid in your application, as I have checked my side and it appears ok. The grid given in the coordinate widget matches that in the MGRS widget and they are both the same as the base mapping in ArcMap.
Just to be sure are you using WGS 1984 Web Mercator as your projection in ArcMap, it should read as srs 102113 in the rest end point. Other than that I'm lost,
Regards
Anthony