The United States changed its official civilian salute to the flag from the raised right hand to placing the right hand over the heart, in response to Nazism. The Nazi salute was in imitation of the original American salute, which we no longer use.
The swastika is an old west Asian symbol, common in Hindu and other religious and artistic designs. The eagle was the symbol of Rome. Hitler borrowed all of his symbols, invented none. The fasces, i.e., a bundle of sticks as an ax handle, was originally Etruscan, and was used by Mussolini for his New Rome before Hitler emerged from the beer hall. Fasces are incorporated into many US regalia, National Guard seal, Congressional mace and flags, etc.
Few identify fasces and eagles with Nazis, but everyone identifies the swastika and the raised arm salute with them