In order to apply for Anna University you need a nativity certificate.

To check the eligibility criteria you can also refer below article

TNEA Details

You can get admission in private colleges through JEE Main.  You can get good private college in other states like: Haryana, UP, West Bengal, Punjab, Rajasthan, Maharashtra. Most of the states have reservation for other state students in private colleges based on JEE Main score/rank. You need to apply separately for them.

Good Luck!!