What is America's claim to the island?

The American claim is based on when Eli Hutchinson Jennings Sr. set foot on the island on October 13, 1856, and raised the American flag. Since then the American position ( or the Jennings' family) has grown in strength since the United Kingdom's position, hence Tokelau's, is progressively getting weak. The western countries have all the recorded history to support its case.
With the establishment of Economic Exclusive Zones (200 miles) around the countries with seashores possession of Swains has become more important to national wealth building.