Another update: CocoonJS is still in the works -- I spent the last few days implementing a "Push to Windows 8 App Store" feature (and some pushing to other app stores as well), though that probably isn't very important to the folks here since C2 already has a nice exporter for that.
What is relevant here is I've updated the plugin to work with Windows 8. Windows 8 has some restrictions where you're not allowed to call external scripts and CSS, so the API and styling for it are stored locally (this is also something necessary for CocoonJS). If you're exporting to Windows 8, change the export mode property for the Clay.io plugin to "Windows 8". It'll do the export and generate a default.html file, in that you'll have to add in:
<link href="/api.css" rel="external" />
It isn't the prettiest solution (having to manually add that in), but it's the best I can do with the current C2.