This doesn’t ring true, I understand the patient wanting to keep his name out of it but where was this? Seems more likely appointments would be canceled ahead of time if care cannot be provided. As far as I know Veteran’s Hospital medical personnel are not subject to furlough, and most VA related things are exempt...though military hospitals and personnel (civilian doctors, nurses, etc.) are subject to sequestration/furloughs.
IIRC, there aren’t civilian doctors and nurses at military hospitals, but it’s been a long time since I was in one.