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Let ${\mathsf D}$ and ${\mathsf H}$ be the self-adjoint, one-dimensionalDirac and Schr\"odinger operators in $L^{2}(\mathbb{R};\mathbb{C}^{2})$ and$L^{2}(\mathbb{R};\mathbb{C})$ respectively. It is well known that, in absenceof an external potential, the two operators are related through the equality${\mathsf D}^2 = ({\mathsf H} + \frac{1}{4}){\mathbb 1}$. We show that such akind of relation also holds in the case of $n$-point singular perturbations:given any self-adjoint realization $\widehat {\mathsf D}$ of the formal sum${\mathsf D}+\sum_{k=1}^{n}\gamma_{k}\delta_{y_{k}}$, we explicitly determinethe self-adjoint realization $\widehat{\mathsf H}$ of ${\mathsf H}{\mathbb1}+\sum_{k=1}^{n}(\alpha_{k}\delta_{y_{k}}+\beta_{k}\delta'_{y_{k}})$ such that${\widehat{\mathsf D}}^2 = \widehat{\mathsf H} + \frac{{\mathbb 1}}{4}$. Thefound correspondence preserves the subclasses of self-adjoint realizationscorresponding to both the local and the separating boundary conditions. Someconnections with supersymmetry are provided. The case of nonlocal boundaryconditions allows the study of the relation ${\mathsf D}^{2}={\mathsfH}+\frac14$ for quantum graphs with (at most) two ends; in particular, thesquare of the extension corresponding to Kirchhoff-type boundary conditions forthe Dirac operator on the graph gives the direct sum of two Schr\"odingeroperators on the same graph, one with the usual Kirchhoff boundary conditionsand the other with a sort of reversed Kirchhoff ones.

${\mathsf D}^2={\mathsf H}+1/4$ with point interactions