I guess it depends on the state. It’s common for local govts. to deny unemployment to those who have been fired. Companies threaten to give bad references if you don’t sign all the agreements they demand.
In Pennsylvania, the employer has to fight it. (Employer makes a statement to the state dept. of labor, was it? The employee makes a statement too, so the employer doesn’t necessarily win).
These days, most companies will confirm former employment, with no additional information.