Firstly you need to understand what red eye is and how it's caused
Red eye in humans is caused by light reflecting off the tapetum lucidium, a layer just below the
retina that is filled with blood vessles. When an intense light such as a camera flash firing occurs
the eyes pupil contracts but too slowly to stop the reflection of the redness of the tapetum.
Red eye is guaranteed to occur with the human eye if the flash is located less than about 2.5 degrees off axis to the eye.
Animals such as cats and dogs have eyes designed to see far better at night than us poor humans, so they have a different physiology of the eye. The 'red eye' and silver 'eye' are actually caused
because they have a far more reflective tapetum lucidium than humans have. The tapetum lucidium in most animals that are nocturnal or have evolved from notcurnal animals reflects light back on to the retina and it's rods and cones thus increasing their night vision sensitivity.
The angle for red/silver eye in animals varies greatly depending on the animal species, the size of the eye and how dilated the pupil is. As a general rule where possible use bounce flash off the ceiling or flash located a foot or more to the side or above the camera to eliminate red eye and silver eye from animal photos.
If bouncing the flash off walls/ceiling etc the flash will take on the colour of the wall/ceiling etc. so make sure it's white!
If you're stuck with an inbuilt camera flash then get someone else to draw the animals attention and shoot profile or 30/60 degrees to the side to eliminate the redeye with animals.