A few moments ago I photographed my cat. My digital camera defaults to the anti-red eye flash setting for indoor shots. When I looked at the image, her eyes were still little pools of light.
I took her picture again with the regular flash setting and there was no difference.
Are the anti-red eye flash settings only functional for humans?
Red eye flashed work by flickering the flash a few times before the main flash goes with the shutter. The idea is that the first few flashes will make the pupil shrink thereby reducing the amount of red-eye. IME dogs (and cats I suppose) have rather large opening/pupils and it doesn’t work as well. Your best bet is one of three options.
1)Post process the image and manually remove the red-eye.
2)Get the flash far away from the camera*
3)Adjust your camera settings and/or brighten up the room so you don’t have to use the flash.
*If you have an SLR camera, this is much easier. Assuming your flash it attached to the camera, you may have some luck with a piece of paper over the flash the deflects the light up against the ceiling. But doing that, may cause the image to come out to dark. There are ways to adjust for that, but it’ll depend on your camera.
What Joey P said. For cameras with a hotshoe (usually SLRs, but I’ve seen some digicams with them), you’d either get a bounce flash that you could point at the cieling (or a wall, or whatever) so the light wouldn’t be going into the subject’s eye before coming back to the camera, or else use an off-camera flash (usually connected to the camera via some sort of cable) to flash from a different angle.
For a long and boring reason the pupil in one of my eyes is always very small - so in photos I tend to have one red eye and one not, which can look odder than two red eyes. The best way I found of dealing with it was to look at a lightbulb for a few seconds… though I suppose that’s not something that’ll help with a cat.
Yeah, all that’s gonna do for ya is leave a big bright afterspot in your field of vision, which will just make it harder for you to take a picture of your cat. 
Even if the pupil of a cat does constrict, you still have the tapetum lucidum to deal with.
Well, if it was a particularly ugly cat, it might help. 
And now i’m going to flee before the MPSIMS-ers attack me for suggesting cats can be uncute.