You can either provide a good lighting when you take a picture so you won't need to use a flash, though you need some cooperation from your pets (like posing or doing certain stuff while the lights are on)
Try to avoid using the flash dead on, point the flash away from their faces. It needs to be at a certain angle and probably you need to play around a little to find the effective angle, I am still working on it.
When it is necessary, there is always photoshop or similar photo editing software, and fixing the evil eyes is one of the simplest operations.