Can my unborn child's father sign over his rights without me agreeing to it. This guy only wants to sign over his rights because of problems with me and him and his girlfriend now who is pregnant also. Will that efect him signing over his rights to my unborn child?

No he can not. He is obligated to help support your child once you give birth and paternity is determined. Signing over his rights is only an option if you choose to marry and your new spouse wants to legally adopt the child.

No he can not just sign over his rights, in fact even if you agreed it is unlikely that the court would allow it.

He does not have to use any right, he does not have to visit, does not have to call or anything

Once the baby is born, file for custody, file for child support and don't listen to his lies, this is common trick for men who don't want to pay child support. Next most likely he will threaten to take the child, or try to get custody, that is often the next threat