Should I have my first child in Argentina

My wife and I have been planning a winter retreat for sometime.  However she just became pregnant and we are debating if this should stop our trip.  She is due sometime in January.  Our options have now become leave by the 6th month and come back after the baby is born.  Also, I'm a US citizen and my wife is a permanent resident.  How difficult or expensive would it  be to give birth in Argentina and would there be any issues with entry while pregnant.

I see your message is a bit old, but if you are still wondering about it I had my first baby here so if there's anything you need to know about citizenship or healthcare feel free to send me a message.

In Argentina medical insurance is fee for everybody.
In that case, you will have not problem but probably you won't be in the nicer hospital that you have probably imagined. For sure if you pay a private hospital you can get both, good care and nice place.
If you are interested in some private medical insurances I would recomend OSDE or Swiss medical.
Best!