They do...
I had to raise my right hand and swear that I renounced my allegiance from my prior country New Zealand BEFORE I was allowed to be naturalized as an American citizen...it was part of the legal process...there is no such thing as dual citizenship in America...
I just don't see that with the majority of immigrants, legal or otherwise anymore, from what I see they want the benefits of being an American without actually being an American........if you know what I mean.