The American Dream is a nationalist ideal proper to the citizens of the United States of America, comprising promises of prosperity and welfare for those who pursue the american citizenship - thus, luring outsiders from developing countries to the land.
In accordance to Barry's mocking description of a typical family's celebration of the Independence Day, he would probably define The American Dream - a somewhat related concept - as "A nationalist sentiment that draws the gullible and desperate to a land of false promises", or something of the sort using the same satirical writing style as in his essay.
