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$\newcommand{\Z}{\mathbf{Z}}$The Chern classes may be viewed as a map $c: \mathrm{BU}(n) \to \prod_{i=1}^n K(\Z,2i)$. Call the target $X_n$. Complex conjugation defines a $\Z/2$-action on $\mathrm{BU}(n)$, whose fixed points are $\mathrm{BO}(n)$. Give $X_n$ the $\Z/2$-action defined by viewing $K(\Z, 2n)$ as $\Omega^\infty(S^{2n,n} \mathrm{H}\Z)$, where $S^{2n,n} = S^{n + n\sigma}$ (here, $S^\sigma$ is the one-point compactification of the sign representation of $\Z/2$), and $\mathrm{H}\Z$ is given the sign action. I think the map $c$ is then $\Z/2$-equivariant, and taking $\Z/2$-fixed points yields the map $\mathrm{BO}(n) \to \prod_{i=1}^n K(\Z/2, i)$ given by the Stiefel-Whitney classes.