Child support for single mother in Califormia

I am from Europe and came to take professional studies at the University. Being in California I have got pregnant and my ex-friend does not want to take any responsibility for our baby.
My first decision was to go home to my country and live with my parents.
But now I started thinking about the delivery of baby in California. I already consultated Child support Centre and they recommended me to stay in California till the birth of baby as it will be much easier to solve paternity problems.
My question is - 1) if I do not have medical insurance covering pregnancy and do not have enough money to pay for the delivery of baby, are there any State hospitals which can give me services with no or low costs?
2) If I deliver my baby in California and later I move to my own country in Europe will I receive any Child allowance, how much and for how long?
Thanks in advance.