Hi! We've moved to Portugal recently, and are a partial match with your situation - my wife is a German citizen and I'm American. As far as my wife is concerned, once we moved into our apartment, we were able to go to the local Freguesia (sort of a local council office) and show her passport, rental agreement, and get an official stamped document that certifies that she is a resident of the town and is entitled to "medical assistance" which apparently enables her to apply for the Portuguese health care number. But, its likely that she will also need an official resident card from the local camara (town hall) which we're going for next.
What has been very clear from the start is that there was no need for her to have paid into any EU system at all (she has been in the US most of her life) and private insurance is not required (though suggested for reasons such as avoiding long waits). You do however have to go through the resident registration process, though that is very straightforward and very cheap (15 euros). By EU law an EU citizen can live or work anywhere with no restrictions. The actual cost of private insurance is a very wide range, and we've used Winsurance, a forum partner to guide us through that process.