Background checks? Maybe if you're joining the secret service. In most cases the employer just needs to check that the qualifications you claim to have are genuine.
Probably you should be allowed to keep quiet about qualifications that you don't want to claim. I read about a case in which a factory worker was sacked with the justification that they had failed to declare their university degrees when applying for the job. They had a degree in politics and the employer believed (probably correctly) that they had got the job specifically in order to be a trade union activist. However, whether that ought to be a valid reason for dismissing an employee is another question.
Amusing story! But in some countries (not sure about the US) it is common to perform a criminal record check, credit check etc. on new employees. In high compliance environments (e.g banks) the checks may span any countries you have resided/worked in for the previous few years.
Probably you should be allowed to keep quiet about qualifications that you don't want to claim. I read about a case in which a factory worker was sacked with the justification that they had failed to declare their university degrees when applying for the job. They had a degree in politics and the employer believed (probably correctly) that they had got the job specifically in order to be a trade union activist. However, whether that ought to be a valid reason for dismissing an employee is another question.