Babysitter was just told another kid in her care has cushing’s syndrome. Should we worry?

No. It's not a common disease. It's not contageous. The diagnosis is very hard to make, even for an endocrinologist. I don't recommend chasing your doc for this diagnosis unless your child has symptoms. See link below to see if you should be worried.
No. This is a problem, usually from the pituitary, in which overproduction of internal steroid causes a variety of systemic problems. It can also present from medical steroid use. It is personal, not transmissable, so your child, or your baby sitter, cannot contract it and it cannot be spread. Do not worry!