The earliest reference for the connection between Stiefel-Whitney classes and Steenrod squares seems to be <A HREF="https://en.wikipedia.org/wiki/Wu_Wenjun">Wenjun Wu</A>, *Classes caractéristiques et i-carrés d'une variété,* C.R. Acad. Sci. Paris **230**, 508–511 (1950), followed a few years later by <A HREF="https://en.wikipedia.org/wiki/René_Thom">René Thom</A>, *Quelques propriétés globales des variétés différentiables*, Comment. Math. Helv. **28**, 17–86 (1954).

Both works are discussed in this <A HREF="http://plms.oxfordjournals.org/content/s3-11/1/741.extract">article:</A>

<IMG SRC="https://ilorentz.org/beenakker/MO/Wu_Thom.png" WIDTH="400">