I have lots of photos of our pet rabbits. Photos of rabbits show extreme redeye, and most redeye correction routines don't work well on them. I discovered that Lightroom CC has a pet eye option. Choose redeye as usual and you will get a panel with options. One is pet eye. I tried it and it worked great. I even tried it on some old photos that had been processed with Picasa. Picasa got rid of the redeye but the result was very unnatural. Lightroom did a good job.
Here is an example of how bad redeye can get on a rabbit. As you can see the room was relatively dark and flash was needed to illuminate the subject.
Here is the same photo after correcting it with Lightroom's pet eye feature.