You have essentially the same rights as a married father but the key difference is that you have to establish them. When married you are presumed to be the legal father. When unmarried you are not presumed.
One area to be very careful about is that you may not be the biological father. If you are convinced you are or if the mother is sincerely seeking to have you be the legal father then go for it. If you have doubts then confer with her about tests that could be taken to prove your paternity. Often times the answer she gives to a paternity question will tell you a lot.
As a dad you should consider assisting where you can as if you were married. Assist economically now. Help purchase items for the child, volunteer to assist in coaching, aim to be present at the birth as a coach, read books about fatherhood, child needs, etc. Try to become as aware as possible of the complex reality of parenthood.
If you sign the birth certificate (assuming she does not oppose it) you are then the presumed father. You then can seek to agree to provide economic support and seek full parental rights through a paternity action. A paternity action establishes your parental rights (in a parenting plan) and your parental responsibilities (in a support order).
Good luck! Parenthood is both a great gift and a great responsibility.
Mr. Kydd has given a very nice answer. As to the health insurance, most NY courts will make a cost/benefit analysis on the issue and assign parental responsibility accordingly. For example, if you already have health insurance that will cover the child, the court would begin by looking to you, but if you can show that the child would get better coverage for a lower price if the child's mother would elect to obtain coverage through her employer, then the court will pressure her (if not order her outright) to do so.
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Once paternity is established, you have potential rights to custody/visitation and obligations for financial support. Speak with a family law attorney in your area for further information.
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