Zev Eleff wrote an article about this story, here. In it he writes:
It is not possible to verify the historicity of the legend. Rabbi Hayim did not record it, nor is the account mentioned by his disciples in the ﬁrst decades of the nineteenth century. Jews in Eastern Europe were vaguely aware of the American Revolution but had little contact with the ﬂedging American Jewish communities and the one thousand women and men who inhabited them.
European Jewry’s awareness of their American counterparts signiﬁcantly increased in the second half of the nineteenth century, long after Rabbi Hayim may or may not have predicted America as an eventual destination for Jewish life and Torah study.
However, he goes on to describe the source as provided by R. Aharon Soloveichik: According to R. Aharon, R. Chaim Volozhiner said this in the presence of his son, R. Itzele, who transmitted it to his son-in-law, the Netziv, who then repeated it at the bar mitzvah of his great grandson, R. Moshe Soloveitchik, father of R. Aharon.