  Let $p\in[1,\infty]$, $q\in(1,\infty)$,
$s\in\mathbb{Z}_+:=\mathbb{N}\cup\{0\}$, and $\alpha\in\mathbb{R}$. In this
article, the authors introduce a reasonable version $\widetilde T$ of the
Calder\'on--Zygmund operator $T$ on
$JN_{(p,q,s)_\alpha}^{\mathrm{con}}(\mathbb{R}^n)$, the special
John--Nirenberg--Campanato space via congruent cubes, which coincides with the
Campanato space $\mathcal{C}_{\alpha,q,s}(\mathbb{R}^n)$ when $p=\infty$. Then
the authors prove that $\widetilde T$ is bounded on
$JN_{(p,q,s)_\alpha}^{\mathrm{con}}(\mathbb{R}^n)$ if and only if, for any
$\gamma\in\mathbb{Z}_+^n$ with $|\gamma|\leq s$, $T^*(x^{\gamma})=0$, which is
a well-known assumption. To this end, the authors find an equivalent version of
this assumption. Moreover, the authors show that $T$ can be extended to a
unique continuous linear operator on the Hardy-kind space
$HK_{(p,q,s)_{\alpha}}^{\mathrm{con}}(\mathbb{R}^n)$, the predual space of
$JN_{(p',q',s)_\alpha}^{\mathrm{con}}(\mathbb{R}^n)$ with
$\frac{1}{p}+\frac{1}{p'}=1=\frac{1}{q}+\frac{1}{q'}$, if and only if, for any
$\gamma\in\mathbb{Z}_+^n$ with $|\gamma|\leq s$, $T^*(x^{\gamma})=0$.
