It depends, make up your eyes with eyes shadow doesn't cause pink eye. However, it can cause eye infection if being shared. And you know what, pink eye is a virus, there are several kinds of pink eye - bacterial, viral, allergy, and reaction to contacts. You may get pink eye from you makeup your eyes with eyes shadow only if you once shared it with someone else who had pink eye.
It will basically not cause the pink eyes by just using the eyeshadows. Although the make up will be full of irritating materials that will stimulate the tear gland, it will not let the eyes get infection like the dust or other dirty things. The pink eyes are a kind of eye disease which is caused by the unhealthy use of eyes or the infection from others with pink eyes. If your eyes don't feel comfortable after using the eyeshadow, your eyes skin must be sensitive.