I doubt Amazon does this. It would expose them to too much legal risk. Almost no large employers (in the US) will say much more than "This person worked here from date X to date Y and they (are/are not) eligible for rehire."

Not being eligible for rehire is a black mark with many potential employers, of course. But then, having Amazon in your work history is also a black mark with some employers.

Nobody is desirable to all potential employers.