I guess it depends on what you mean by business! The thing is, “business” is just a catch-all term for doing things - a business is simply an org that provides goods or services. There’s management, finance, accounting, HR, marketing, advertising, etc., in every kind of organization. Management is probably the easiest to combine with health care provision, since most health care providers who stay in the business for any length of time end up managing people to a certain extent.
As you may know, currently many allied health care providers actually have their own private practices which they run. Physicians, dentists, optometrists, podiatrists, physical therapists, occupational therapists, and even some nurse practitioners can begin their own practices or clinics and are responsible for patient care and running their practice. The landscape is changing; many practices are being purchased by hospitals and healthcare management organizations, but many allied health providers are responsible for managing their business and also caring for their patients.
Pharmacists, too, usually have one hand in each pot: they do counseling and advice for their patients on how to take medication, but are often involved in the day-to-day business of managing the pharmacy. This is especially true if you choose to start your own private pharmacy (as opposed to working for Walgreens or something).
Many healthcare providers take their patient care expertise into business. You usually do need some years caring for patients first, but after you have put in some time, you can move into management. Physicians and nurses can move into the administration and management of patient care at hospitals; they can direct patient care plans, oversee other practitioners, and make executive decisions for the unit at the hospital. Many physicians and nurses move up the ranks - each hospital has a Director of Nursing, for example. Nurses can become a charge nurse after as little as 3 years of experience depending on the hospital.
Many healthcare professionals (mainly physicians, physician assistants, nurses/nurse practitioners, and dentists) go on to work as consultants for private companies, like insurance firms and strategy consulting agencies. At this point you usually aren’t seeing patients, though; you use your expertise to decide what a health insurance agency might cover as medically necessary, for example, or your background to help healthcare organizations restructure or decide new areas to launch into.
Hospital administrators don’t see patients. Many healthcare professionals cross over from patient care to full-time administration (as opposed to managing a health unit, in which role they would manage others while still seeing patients).
If you want to provide care to patients, you need to major in something that will get you licensed to do that, or get a master’s degree. For example, if you wanted to be a physician, you can major in anything, take the pre-med prerequisites, and go to med school. Optometry and dental school work like this too. If you want to be a nurse, you need to major in nursing in undergrad, or you can major in something else and go to an accelerated BSN or a direct-entry MSN program (usually a 2-3 year program in which you get both your RN/BSN and your MSN, and can enter practice as a nurse practitioner) after college. Pharmacy school works a bit different; you technically can enter a PharmD program after 2 years of college, but I think most admitted students have a BA.
If you mix the two academically that would probably be done on the graduate level. You can, for example, get an MD/MBA or an MSN/MBA. There are also several MSN programs in nursing leadership or nursing management.