People from Barbados in the USA?

I asked this before but I guess the person answering this thought I was referring about myself, which I am NOT. Therefore, If someones great-grandparents were born in Barbados, but their grandparents, parents, and the person theirself was born in the USA, can that person say their African-American instead of Barbadian, Bajan, or simply black? So make a long story short, the persons ENTIRE family is of Barbadian descent, however, the person, their parents, and grandparents were born in the USA. It was only the great-grandparents on ALL sides that were born in Barbados then came here.