If you want to take a picture without the flash reflecting, you'll either have to turn off the flash or dont aim the camera directly on her so the flash wont directly hit her eyes and come back. The flash reflection you are seeing is not like red eye, which has a solution. Red eye occurs when a flash goes off so fast that your pupils dont have time to close enough. The red is actually the blood in the back of the eye and retina. Thats why most cameras have that double or triple flash before actually taking a picture, because it gives your pupils more time to close. (Dogs can get red eye too...but the same thing fixes it.) All canines (and cats and lots of other animals) have what is called a tapetum lucidum. These animals evolved to hunt at dusk and night time when there are low light levels. The tapetum is a reflective layer in the back of the eye that unabsorbed reflects light back so it has another chance of being absorbed, therefore increasing the light seen by the eye and seeing a better picture of the bunny hopping through the grass! We don't see it in pictures of humans because we don't have them.
This from the graduate student who just wrote a paper on vision in dogs....
Hope it helps!