What credentials should an eye surgeon have to perform eye muscle correction?

There are many muscles around the eye, including those that move the eye and move the eyelids. Your question suggests that you are referring to muscles that move the eye. These six muscles, called extraocular eye muscles, are usually operated on to treat conditions such as abnormal alignment of the eyes or strabismus. Most ophthalmologists who are board certified by the American Board of Ophthalmology are qualified with the proper credentials to operate on these muscles. Some ophthalmologists may choose not to perform this surgery, however. Some forms of strabismus are complicated and more difficult. These may be referred to an ophthalmologist specializing in pediatric (and strabismus) ophthalmology. These ophthalmologists have had extra training in pediatric ophthalmology (where most cases of strabismus are seen) or have limited their practice to such surgery. However, there are no special credentials for these board-certified ophthalmologists to limit one's practice to these areas. I would encourage looking for an ophthalmologist that is board certified.