So your wife/fiance/girlfriend will be having the baby in Warsaw?
My wife will be having our second child in Warsaw around February.
She (or he) will be a Polish citizen and you would have to get her a US citizenship if you plan on it.
Since neither I nor my wife are Polish, my understanding is that our child will be born an American citizen by blood and we will just have to register his or her birth with the US Embassy. In our situation our child would only be born Polish if he or she would otherwise be stateless, which doesn't apply here. We will need to get a passport.
If anybody wants a copy of that, let me know: I can certainly scan and email it (I can probably supply a text-only version if needed).
I would love to see that article. I'll PM you my email address.
As far as insurance, at this point medicover looks like our best bet. I'm not dead set against using state hospitals but I can't get a visa for my family without proof of health insurance, and medicover would cost less than half what insurance from one of the foreign companies that caters to expatriates would charge. Medicover has English-speaking staff which helps too since my Polish isn't very good yet. I don't think there is a way for non-EU citizens to use the state system (If I'm wrong please let me know, I'd like to know all my options- the main concern is the visa thing).