Your Chao can have a variety of different faces. You can see the different eyes and mouths possible below. (inspired by Shining Chaos Chao)
Normal Eyes
Half-Open/Tired Eyes
Evil Eyes
Happy Eyes
No Mouth
Happy Smile
Sad Mouth
Evil Grin
Angry Evil Frown
Tired Mouth
It is random which face your Chao will receive.
However it seems to be influenced by the faces of the parents (if egg was recieved via mating)
and also strongly influenced by the face of its previous life (if egg was from a reincarnated Chao).
On the Gamecube version, as soon as the egg is in the garden, its face will be randomly set.
Once the egg is in the garden there is no way of changing its face (unless you reset the console without it saving).
On the Dreamcast version of the game, you can hatch it anytime and the face will be randomised.
Note that it has been proven that the way you hatch the egg has NO effect on the Chao's facial appearance.