Is there a job that a 14 year old person can do with health care?
I don't think there is much direct care that a 14yo can do, but most hospitals have volunteer departments that allow you to get on the floors and interact directly with the patients. The longer you are there and the better the nurses know you, the more they will ask you to do. If the hospitals don't want you the nursing homes surely would, if only to read to people, or man the dayroom or kitchen and help the residents.
You may be able to get parttime jobs in the kitchen, or maintenance department, but probably wouldn't have much patient contact.
When you are older try taking a CNA course,
from my experiences it pays to know people. if i were in your position i would start learning and taking classes to prepare for the courses that will certify u later. and talk to people about it, show interest and make friends with them. all of the great jobs i have had are cuz i got hooked up by someone.