Do I need to be worried if an employer tells me that they want to hire me and asks for references I can't provide?

[bluescoop, BSN, RN]

I just got an email congratulating me for a job I applied for and saying that I am their preferred candidate. To facilitate employment process, they asked for me to send them 3 references who they will call, and they asked that the references be from supervisors if possible. I don't have any supervisors who gave me permission to use them as a reference and I was fired from my last job because of my supervisor thinking I was a bad nurse. I just sent them the contact info for a DON at a SNF I worked at years ago. The DON didn't give me permission to use her as a reference, but she had told me I was a good nurse and she is the only supervisor I've worked with who ever said something positive about me. I'm just afraid they will rescind the employment offer if the reference checks don't go well. I can think of a few other colleagues I could use for reference, but I am very limited in terms of professional connections from supervisors/managers. I don't want them to take back the employment offer so I need to know how important the references are. I am not understanding why they are asking me to send them references *after* a conditional offer of employment has already been made. I would think they would want to check references before selecting a preferred candidate if it's that important. Does anyone have experience with HR and how the hiring process normally proceeds? This is for an RN position at a community health clinic.