I use the magic wand tool function in PS, but I use Photoshop 7.0. Because Elements is an inexpensive strip-down tool, it doesn't have a quick easy method for animal green-eye removal.
You can highlight it -- selecting only the affected green-eye area. Then perhaps desaturate the color (removing all tint or hint of color). Be aware that you might need to darken exposure of that selected area too.
Even better process is the red-eye removal tools in Paintshop Pro 7. They have one for human red-eye and animal green-eye.
Oh, wait..why not just send me the file. I'll fix it for you! I do this professionally.