I don't know how it is in Europe, but those sorts of things are not typically covered by insurance in the US, except in cases where the reconstruction is related to a medical condition or accident (I got a nicer nose after breaking the original in a bicycling accident; the reconstruction was covered by health insurance). I assume those coming from EU to US for treatment would be self-pay for the treatment, yes?
I have no idea how payment would be arranged. It's possible that if a country's national plan covers something but no providers are in-country, they would cover the cost.
Some hospitals in the US are free for everybody (some research hospitals and hospitals for sick kids) and I assume that would extend to Europeans.