Perhaps the author wanted to remind us of how what we think of as "American" culture (even "democracy') has always been a process of different ethnic groups mingling; there is no such thing as "purity" in human history. And back in the late 19th century, immigrants from Italy, Greece, and Eastern Europe were in many ways reviled as much as those from Hispanic cultures are today. Yet look at how much these groups contributed to what we consider "American" now. If that was her purpose, using a family with a Greek background makes good sense. It also makes for a stronger parallel with the history of Vineland, and a house (symbolic of our nation) that was once predominantly "Anglo" and is now multicultural.
I think Kingsolver also wanted us to see this family as typical Americans in having a mixed heritage, people who have made some progress towards the "American dream," who have certain expectations and assumptions that were once reasonable -- such as two professionals being able to afford a decent home -- but which now appear out of reach. They resent to some extent both the privilege of the new financial elite, and the relative success of new immigrants in their neighborhood, whose ways are different from their own. It takes Tig -- Antigone -- to be the moral compass, to see clearly that we are all in the same boat (Mother Earthship, so to speak).
Thanks for raising this question!