My child's father is coming out of prison and has never been there for our son, now he wants to have visitation rights, is he entitled to them or do I have full custody to where he would not be able to take him?

How old is your son? Depending on how far he wants to take it in court he may very well have rights to visitation.

Do you have a reason to not allow him to visit his son?

He has the right to file in court to get other rights, such as shared custody, visitation and other rights

Not being on the birth certificate does not take away his rights. It just means he has to go through the courts to enforce them. He will need to go to court to establish paternity. He will need to show he is not a danger to his children.

But unless you can prove he will be, he is likely to get visitation.