If your camera has "red-eye reduction" that should help. It flashes twice and takes the photo on the second flash. The first flash helps to contract the pupil so you won't get red eye. Usually if you're looking for it on your camera it looks like and eye with a line through it.

Also if you take the photo in a brighter area - like in the daytime near a window or outdoors - then even if you use the flash you shouldn't have the problem as much, because the dog's pupils will be smaller and the flash won't reflect off the back of the eye as bad.

Hope this helps!