Is it illegal for me to ask a prospective employer to record the reference call to verify that the former employer is not stating untrue facts that they shouldn't be?
It is not illegal for you to ASK, but depending on your state law it might be illegal for the prospective employer to do so. Also, if an applicant asked me to record a reference call in case the employer was lying, that would send up dozens of red flags. I do NOT recommend that you make such a request.
Also, what is the former employer allowed, and not allowed to say, and how?
The employer may say anything that is true, or that they honestly believe to be true. They may not say anything that is false or misleading. They may not say anything that relates to any medical information they may have on you.
would the former employer have to know that the call was being recorded in order for it to be admissable in court if the former employer tries to "blacklist" me?
This depends on state law. Also, your use of the term blacklist is incorrect. A negative reference is not a blacklisting - that term would refer to your former employer calling up possible employers, unasked and uninvited, and encouraging them not to hire you.
FYI, after 25 years in the HR and benefit business, I have NEVER had an employer do that, nor has any HR manager that I know.