Can't figure out how to properly rig flat eyes

I should start out by saying while I’ve been using Blender for a while, I’m only just now learning about rigging and weigh mapping in it, so if this is an obvious answer, have mercy.

So, the model I’m trying to use isn’t one I made myself. It’s actually something imported from another program called MMD made possible only with an addon. I mention that because the rig itself may only work properly for said program, so there is one problem I’m having with it. It’s simply something I’ve been playing around with. Everything is fine with it; it moves properly, the physics work properly, but the eyes are giving me trouble. They’re simple irises rigged to simply slide left to right, up and down, to simulate eye movement. It works fine in the other program, and it does at first for Blender…

…until I actually move the rig, in which case it looks like this.

Before moving armature

After moving armature

The eyes rotate awkwardly to wear they aren’t properly in the place they should be. I’ve been looking up tutorials and instructions on other posts on here, but nothing I’ve tried seems to work. If someone could take a look at the blend file and see what’s up, I’d greatly appreciate it. Again, I’m not very good with rigging things.

http://www.pasteall.org/blend/35586

Here’s something I remember seeing not all that long ago on rigging 2D eyes using cameras and the UV-project modifier. Not sure if the original MMD rig conversion is done that way, you may have to custom re-map the eye part of the mesh and redo the textures for them to allow the UV coordinate offset.

[video]https://youtu.be/cCY99_HqVHI[/video]