For instance, you would expect separate changing rooms and showers for men and women in public facilities. Since medical treatment involves touch and sometimes nudity, it's not that surprising that some people could regard this as another area where the gender of the nurse (or doctor) matters. Remember that it's not just Muslims who feel that gender makes a difference:
In general, the rule is to have a “chaperon” present to protect patients against possible sexual misconduct, and to make them feel more secure during intimate examinations. The official AMA guidelines state, “From the standpoint of ethics and prudence, the protocol of having chaperons available on a consistent basis for patient examinations is recommended.”
In fact, most institutions mandate that chaperons be present for rectal, pelvic and breast exams. Whether this actually happens is another story.
In practice, adherence to these rules varies widely. Often the issue is not having extra staff available, or not having enough time to track someone down. Most male doctors are assiduous about having a chaperon when performing a breast or pelvic exam on a female patient, though many do not use chaperons for male patients.
But female doctors, as a group, rarely use chaperons for either male or female patients during genital exams. In some studies, the number was as low as 10 percent.
http://well.blogs.nytimes.com/2010/12/02/do-patients-need-a-chaperone/#So this really is very different from something about race. As drm604 says, it'd be interesting to know if the women themselves were demanding a female nurse, or if it really was just "the clinic's male Muslim clientele" who were trying to control things (which they have no right to).