If a child has chicken pox, can she still fly on an airliner?

Not if contagious! Chicken pox is highly contagious and is spread very easily in closed in areas. The virus will be recirculated in the a/c system. So no.
Chicken pox. Usually chicken pox is contagious for the first week that they are sick. The actual rash doesn't show up for the first 2 days, so usually you are walking around feeling a little sick, but don't know you are contagious with chicken pox because you haven't broken out yet. Once the pox are scabs, then you're not contagious any more.
I would not... ...Bring a child with chickenpox on an airplane. She could infect any passenger who hasn't had it or been vaccinated against it. More importantly, a very elderly passenger, or one with a defective immune system, could develop chickenpox pneumonia, which is potentially fatal. I would not take a chance like that if she were my child.