To not bat any eye means to not display any sort of emotional response, which is what the people he hired would do at the slimy schemes.
Consider the staetement "He hired the kind of people who would not display even a hint of an emotional response such as consternation at the slimy schemes.". Based on this statement, is "He hired the kind of people who would not bat an eyelash at the slimy schemes." a true statement? Yes/no/unknown