Dear aspirant hope you are doing well.

Yes, students from private  university or college are eligible for the  government exam on the completion of their degree . Candidate can apply for the SSC CGL, SSC MTS, UPSC, DRDO, RAILWAY EXAMINATION, SSC CHSL. A candidate must fulfill the eligibility criteria for the exam. It doesn't matter the university is public/private but it must be affiliated from the relevant authorities.

Hope it's helpful for you, thank you