Bless you, I know exactly how you feel as i've had pretty much the same thing to me!
When I used to do hair mobile, I had to cancel a friends daughters hair appointment as my 15 month daughter was taken into hospital in the middle of the night.
The abuse I got was unreal, she slagged me off all over fb and to mutual friends. All because she was going out on the Saturday night (I cancelled on the Tuesday morning for the wednesday)
Best bit was....she was the receptionist on the children's ward my daughter was on so she knew just how poorly she was, took all my restraint not to throttle her
I chalked it up to experience and knew that my children will always come first, she needed me more than some petulant 20 year old who was throwing her toys out of the pram. As you said, if it was the other way round you'd be sympathetic and understanding. I'd never hold it against my hairdresser if she had to cancel or rearrange an appointment because her child was ill.
God, some people are so selfish! Are they that vain that their looks come before poorly children?! Honestly, I am speechless they could bad mouth you!