I just gave birth to my first child at Yale-New Haven hospital in April and we have the Yale Health Plan insurance. As far as I know the health plan will not cover the cost of a birth anywhere except Yale-New Haven hospital, so your options may be limited to that if you can't afford to pay for a birth somewhere else. That being said I had a wonderful experience at Yale-New Haven. The nurses were great and very supportive of my decision to have a natural childbirth. They offer birthing balls, rockers, and showers in each of the rooms and the nurses suggested different positions and techniques for me to try during labor. They seemed very excited to have a mom who didn't want an epidural.
The only negative thing I would say about the health plan is that there are a bunch of doctors and midwives in the practice and you really have no idea who will deliver your baby. They try to rotate you through all of the delivering doctors in your last trimester so that you will meet them all, but I had never met the doctor that delivered our son. I don't have any regrets though, she was great and I had a very good birth experience. Because of that unknown I would make sure you have a birth plan. They asked for mine as soon as I got there and made copies so that everyone knew what I wanted and it was kept in my file. I really felt like the doctor and nurses did their very best to follow my plan. The only thing that did not go as planned was that my fluid was meconium stained, so my son went right to the pediatrician instead of to my chest when he was born. But that was obviously in his best interests.
I hope that information helps. Good luck with your birth!