Even better process is the red-eye removal tools in Paintshop Pro 7. They have one for human red-eye and animal green-eye.
That's exactly what I used. The hardest part was getting the targeting right so the dog doesn't look cross-eyed or doped up.
Here's a link to the latest version of the software
if you want to give it a try. They have a 30 day trial on the full download.
I have not tried anything past Paintshop Pro 7 I purchased more than 5 years ago. I am still very satisfied with what it does. Therefore, I can't vouch for the software after Corel took it over, but I still read good things about it. I had learned Photoshop at work and that knowledge made learning the similarly organized Paintshop Pro very easy.