You have two problems. The first is that an animated gif has no special status in DVD authoring. It is an anmiation. How do you get animations into a DVD project? You import them as movies - either as assets that get puit on timelines, or assets that become part of a motion background.
Yes, Encore has a special feature of making animated thumbnails, but this is just a fancy way of putting thumbnail-sized bits of video into a video background of a menu.
I think you resolve the animated gif issue if you convert it to a video format Encore understands.
http://helpx.adobe.com/encore/using/importing-assets.html#supported_fi le_formats_for_import
The second issue is that you want a button highlight to be an animation. Also not part of the DVD spec or what Encore will do. You can fake this, by having a button autoactivate and to go to a different menu that has the animation for that button as part of the background. You create as many menus as you have buttons with that characteristic.
Note that a mouseover won't autoactivate, so this doesn't work on a computer the way solme users would like.
Scripting is a way some authoring applications can handle such things, but Encore does not provide scripting.