Your Chao can have a variety of different faces. You can see the different eyes and mouths possible below. Click on the links to update the image.
Your Chao will start with an entirely random face, and it is impossible to change the Chao's face during its life, although it can change through reincarnation. The face that it gets seems to be influenced by the faces of its parents (if the egg was recieved via mating) and also strongly influenced by the face of its previous life (if egg was from a reincarnated Chao).
The face is determined from the very moment that the egg is created or purchased. This means there is no way of changing its face once the egg exists. The only way of getting a different face is to reset the console without saving, and obtain the egg again. If you have already saved with the egg in your Chao Garden, then there's nothing you can do about it.
Please note that while many people believe otherwise, it has been proven that the way you hatch the egg has no effect on the Chao's facial appearance.