I'm American. My family came here a couple of generations ago to be Americans. They learned English and dropped the other language because they came here to be Americans. It's not that they're not proud of where they came from but where they were isn't where they wanted to be. America is.
Immigration without assimilation is occupation. "Irish-American" is as lame to me as "African-American". Go to another country and I seriously doubt anyone there would claim to be "American-Irish" or "American-African".
That's nonsense, borderline PC garbage.
Assimilation is a two way street. A large part of the american identity, especially regional ones have been defined by the immigrant groups that settled them.
Being proud to be irish, isn't simply having some emotional connection to some foggy island, it's about acknowledging the role the irish have played and still do in creating north america.
Without the irish, germans, scots, Italians and blacks america would be a very different place.