(AnemoneLabSNC has picked this extension up and currently holds the latest version. I'm leaving this post up for historical purposes.)
Here's my 30 minute crack at this doing exactly as I said: Wrapping spinehx into a plain extension (attached) and writing a little bit of code (based off their sample) to use it.
The assets are dunked into the "extras" folder as described here.
http://community.stencyl.com/index.php/topic,24729.0.html
All the code I used is housed inside the Spine Animation behavior. All it does is import the skeleton and mount it to the actor via addChild(). Someone with some motivation to repackage this could turn it into a nice, reusable behavior that can support multiple animations and anything beyond the barebones I have here.
Instructions for the Lazy
(Works on all platforms since it's pure Haxe.)
1) Import the Spine extension (unzip spine.zip, copy into [STENCYL]/plaf/haxe/extensions).
2) Import the project (SpineTest.stencyl). I'd double-check that the Spine extension is enabled after you open it.
3) Run. Works on all platforms.
4) If you want to tinker with it, read what I said above.