I have tried using a speedlight on animals and birds and end up with the equivalent of redeye. However, with non humans, it seems to always be a silver color. When I try to use redeye reduction in post processing, I get a response of "redeye? redeye? what redeye?". Is there any post processing software that will respond to colors other than red?
In animals its often referred to as SteelEye. And it is very difficult to fix. Unlike redeye, where you can remove saturation of the red channel, SteelEye affects luminescence and extends broadly across the eye.
The obvious solution is to not use flash when photographing animals susceptible to steeleye. You might also be able to use off camera flash but it would have to be well off axis. And in some cases you can simply reduce the flash to a mild fill and steeleye will be prevented. The answer varies by subject. My usual approach is to not use flash or use a minimal fill flash.
In post processing, the options are tougher. You can clone in an eye from another image. You can also clone out the gray steeleye. It is also possible to darken the eye, but I find that approach can look unnatural.
The other approach in post processing deals with blue color in steeleye. In that case you might be able to desaturate the blue tone in the eye and return it to a more neutral black and white tone.
In addition to Eric's comments, you might also try using the color replacement tool to try and restore the natural iris color. I have found it often takes a combination of techniques to resolve this problem.
For some reason being off camera on a bracket is often not enough. I think you need to be well off camera - several feet or even a remote flash. That provides directional light that does not reflect back to the camera. This is very subject specific and is based on the specific characteristics of the subject's eye.