Babysitter wont watch 2 year old because of a rash? Is she overreacting?

I wouldn't think this is her overreacting, just her way of handling something she does not understand. If it were me and I was not a long time friend or family member of yours, I would be cautious of any signs of a medical problem. It's always better safe then sorry. She just wants to be sure that nothing serious is wrong and certainly doesn't want to be responsible if something happened while in her care.