It seems pretty obvious that gentiles shouldn't make Berachoth on any Mitzvoth (their 7 aren't exactly ritualistic, and they were not commanded in the rest of ours). But are there any Berachoth that they may/should say? Perhaps Nehenin (foods, scents) or Shevah (praises)? Maybe Shehehiyanu on certain occasions? Maybe even a judge who presides over a case (since they are commanded to have a justice system, which he is implementing)?
We are instructed to answer Amen after a Bracha, if it's said by a Jew. If we hear the whole Bracha from a goy we also answer, otherwise we don't (we're afraid he said things we don't agree with) [SA & Rma OC 215:2]
So, apparently, a goy can say a Bracha, and we even say Amen. (Rma says
ועונין אמן, I'm not sure if this should be understood as an obligation or not. I tend to understand that it is and obligation)