Here are some more historical references confirming that the sentiment over the years since the founding was you are either coming here to be one of us or don't come here.
Theodore Roosevelt's ideas on Immigrants and being an AMERICAN in 1907.
"In the first place, we should insist that if the immigrant who comes here in good faith becomes an American and assimilates himself to us, he shall be treated on an exact equality with everyone else, for it is an outrage to discriminate against any such man because of creed, or birthplace, or origin. But this is predicated upon the person's becoming in every facet an American, and nothing but an American...There can be no divided allegiance here. Any man who says he is an American, but something else also, isn't an American at all. We have room for but one flag, the American flag... We have room for but one language here, and that is the English language... and we have room for but one sole loyalty and that is a loyalty to the American people."
Theodore Roosevelt 1907
"I am not a Virginian, but an American." --Patrick Henry
Get the gist? Be an American, not an African-American, Hispanic-American, Euro-American, Asian-American, not some other hyphenated American. Be an American, or get out."The name of American...must always exalt the just pride of Patriotism, more than any appellation derived from local discriminations." --George Washington