The Yugoslavian project was a Serb lead project, that sought to unite the various Southern Slavic peoples under Yugoslavia. The united Yugoslavia was invented by Serb nationalists like Ilija Garasanin in the 19. century, and came to fruition after WW1, when the former Habsburg lands of Slovenia, Croatia and Bosnia came under Serb governance in the Kingdom of Southern Slavs, or the Kingdom of Yugoslavia.

For Serb nationalists it seemed obvious that these peoples belonged together, as king Stefan Dusan had united these lands in a short lived empire in the 13. century, that helped form the national identity of Serbs. However by the 20. century, Serbs had longlasting cultural and diplomatic ties to equally orthodox Imperial Russia, while Slovenes and Croats esspecially had been under Habsburg hegemony for centuries. Slovenes and Croats was therefore to some extend "Germanized" and primarily catholic. Some Slovene and Croat nationalists hoped to establish their own nations after WW1, even though the majority to some extend, recognized that they lost the war, and as consequence were to be absorbed into a greater Serbia. So from the onset of Yugoslavia, there were dissatisfaction with Yugoslavia among Slovenes and Croats, amplified by the government consisting of primarily Serbs and some Serb friendly minorities.

When WW2 came to Yugoslavia, the Croat Usashe esspecially showed that Croat nationalism was alive, and their crimes against humanity, helped plant the seeds of some aspects of the civil war that came in the 1990's. However after WW2 when the Ustashe was eliminated, and Tito grapped power in Yugoslavia, the Yugoslav project began in earnest. Public schools taught the various nationalities of the history and achievements of the other nationalities, which helped form a new generation of people, who to some extend began to consider themselves as Yugoslavs instead of as Slovenes, Croats, Serbs etc. However ones nationality remained an integral part for many Yugoslavs, and the Yugoslav cultural project never saw complete fruition despite decades of attempts.

So to answer your question about Yugoslavs coming to America. It really does depend on the individual whether he sees himself as a Yugoslav-American or nationality-American.

Source:

Bennett, Christopher. 1995. Yugoslavia’s Bloody Collapse: Causes, Course and Consequences. London: Hurst & Co.

Mojzes, Paul. 1995. Yugoslavian Inferno. New York: Continuum.