I recently received a message from my estranged mother. She says there has been a diagnosis in my family that could affect my daughter and/or myself. She refuses to pass on relevant info as to what it is. Is there any way I can force her to tell me what it is so I can take the necessary steps to get my daughter and myself tested.

Nope... not really. She doesn't have to tell anyone anything. In fact one spouse legally isn't required to share that information with their other spouse. Its purely voluntary, and considered privileged and very private information from a legal aspect.

Not really, but my guess she is just messing with you, she would not have called unless she wanted to make you worry. I am sure she is estranged for a reason, perhaps keeping no contact will be best