The magnetic properties of the $S=1/2$ two-dimensional square-lattice antiferromagnets ${\mathrm{Sr}}_{2}{\mathrm{NiO}}_{3}X$ $(X=\mathrm{Cl}, \mathrm{F})$ with the trivalent nickel ions in a low-spin state were studied by magnetic susceptibility, heat capacity, neutron powder diffraction, high-field electron spin resonance (ESR), muon spin rotation and relaxation $({\ensuremath{\mu}}^{+}\mathrm{SR})$ measurements, and density functional theory (DFT) calculations. Both oxyhalides are isostructural to an ideal quantum square-lattice antiferromagnet ${\mathrm{Sr}}_{2}{\mathrm{CuO}}_{2}{\mathrm{Cl}}_{2}$, but the chlorine/fluorine anion exclusively occupies an apical site in an ordered/disordered manner with an oxygen anion, resulting in the formation of highly distorted ${\mathrm{NiO}}_{5}X$ octahedra with an off-center nickel ion. Magnetic susceptibility measurements revealed a remarkable difference between these two compounds: the magnetic susceptibility of ${\mathrm{Sr}}_{2}{\mathrm{NiO}}_{3}\mathrm{Cl}$ exhibited a broad maximum at approximately 35 K, which is typical of low-dimensional antiferromagnetic behavior. In contrast, the magnetic susceptibility of ${\mathrm{Sr}}_{2}{\mathrm{NiO}}_{3}\mathrm{F}$ exhibited spin-glass-like behavior below 12 K. No anomaly associated with long-range magnetic ordering was observed in the heat capacity, ESR, and neutron powder diffraction experiments. However, ${\ensuremath{\mu}}^{+}\mathrm{SR}$ measurements revealed the emergence of a static magnetic ordered state below ${T}_{\mathrm{N}}=28\phantom{\rule{0.28em}{0ex}}\mathrm{K}$ in ${\mathrm{Sr}}_{2}{\mathrm{NiO}}_{3}\mathrm{Cl}$ and a short-range magnetic state below ${T}_{\mathrm{N}}=18\phantom{\rule{0.28em}{0ex}}\mathrm{K}$ in ${\mathrm{Sr}}_{2}{\mathrm{NiO}}_{3}\mathrm{F}$. The DFT calculations suggested that the unpaired electron occupied a ${d}_{3{z}^{2}\ensuremath{-}{r}^{2}}$ orbital, and ferromagnetic couplings between the nearest-neighbor nickel spins were energetically favored. The mechanism of ferromagnetic superexchange interactions and the reason for the difference between the magnetic ground states in these nickel oxyhalides are discussed.