"Contractor pre-screening questionnaire"

Can a private company contractor require subcontractor’s employees to answer very personal questions to determine eligibility of clearance?

If the subcontractor needs a clearance, yes.

Companies can ask pre-employment questions but cannot discriminate against you in areas like sexual orientation, gender, disability, ethnic origin, marital status, etc… If they ask questions about drug use, criminal history, alcohol use, and previous employment disciplinary actions they must apply an established criteria/policy fairly and impartially across the board to all applicants, otherwise they would be in violation of EEOC laws.