Need advice on having a conversation with a hospital about feeling black balled ? I was fired (10yrs ago). Would love for them to get over it & not hold me back as I have to put them in my resume. How do I get them to remove “ineligible for rehire”?

This is a sticky question. Are you applying with the hospital or other facilities in the same company? The reason I ask is most hospitals only report that you worked from this date to that date. However, it also matters what you were terminated for at the time. If the hospital uses a outside company like Group 1, you can reach out to them as well to find out their request process to help.Normally, after this amount of time, you can write the Director or VP of HR asking for them to review the ruling. Send a "I was young and sorry for the event letter". Explain how that event has effected your current goals and made you a better more aware professional. Include a copy of your resume to show your career.
Hopefully, you have had some lengthy stays at other positions after you no longer worked for them.
Best of luck.

If you want to work for that institution you are going to have a tough time. However, typical background checks look at years of employment. If, you are denied employment based on a prior employer comments you can take them to court for slander.

10 years seems to be a long time to worry about it. If your work history is good and you have some good references don't worry about it.