  Let ${\mathsf D}$ and ${\mathsf H}$ be the self-adjoint, one-dimensional
Dirac 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 absence
of 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 a
kind 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 determine
the self-adjoint realization $\widehat{\mathsf H}$ of ${\mathsf H}{\mathbb
1}+\sum_{k=1}^{n}(\alpha_{k}\delta_{y_{k}}+\beta_{k}\delta'_{y_{k}})$ such that
${\widehat{\mathsf D}}^2 = \widehat{\mathsf H} + \frac{{\mathbb 1}}{4}$. The
found correspondence preserves the subclasses of self-adjoint realizations
corresponding to both the local and the separating boundary conditions. Some
connections with supersymmetry are provided. The case of nonlocal boundary
conditions allows the study of the relation ${\mathsf D}^{2}={\mathsf
H}+\frac14$ for quantum graphs with (at most) two ends; in particular, the
square of the extension corresponding to Kirchhoff-type boundary conditions for
the Dirac operator on the graph gives the direct sum of two Schr\"odinger
operators on the same graph, one with the usual Kirchhoff boundary conditions
and the other with a sort of reversed Kirchhoff ones.
