The earliest reference for the connection between Stiefel–Whitney classes and Steenrod squares seems to be Wen-tsün Wu, Classes caractéristiques et i-carrés d'une variété, C.R. Acad. Sci. Paris 230, 508–511 (1950), followed a few years later by René Thom, Quelques propriétés globales des variétés différentiables, Comment. Math. Helv. 28, 17–86 (1954).
Both works are discussed in this article:
