Let K be a complete, algebraically closed non-Archimedean valued field, and let \(\varphi (z) \in K(z)\) have degree two. We describe the crucial set of \(\varphi \) in terms of the multipliers of \(\varphi \) at the classical fixed points, and use this to show that the crucial set determines a stratification of the moduli space \(\mathcal {M}_2(K)\) related to the reduction type of \(\varphi \). We apply this to settle a special case of a conjecture of Hsia regarding the density of repelling periodic points in the classical non-Archimedean Julia set.