Calls even after told it's a wrong number.
Appears to be a collection agency

Annoying calls early in the AM.
Appears to be a collection agency.

Seemed to be a (vaguely) legit call about health care bills

The company is a healthcare revenue cycle management company that works for hospitals and physician offices around the country.  Very professional.  They are responsible for managing the billing operations and patient communications processes for many large hospital systems and doctor offices.