The person's insurance company will provide insurance benefits whether the clinic is open or not.

This reminds me of the insurance program at my university, in which all claims (medical problems) that occurred during the hours that the clinic was open had to be dealt with there first.
If something happened to you outside of clinic open hours, however, you were covered for any medical attention you had to seek out.

Health Maintenance Organizations, or HMOs, also work this way in the United States. All treatment should start at the clinic, but if you get hit by a car at 2AM you'd be covered for that too.

Hope this helps!

It seems to me that this is a similar arrangement to the German Notarzt system. While the clinic/office may be closed, you can always reach a physician outside of normal office hours. Whether or not this is only for medical emergencies depends upon the context.