The temperature dependences of the dielectric properties at different fixed pressures, and the antiferroelectric phase transitions of squaric acid ${\mathrm{H}}_{2}$${\mathrm{C}}_{4}$${\mathrm{O}}_{4}$ and its deuterated form ${\mathrm{D}}_{2}$${\mathrm{C}}_{4}$${\mathrm{O}}_{4}$ have been investigated. Using the four-sublattice pseudospin cluster Hamiltonian, together with a pseudospin-phonon interaction term, the statistical Green's-function technique has been applied to explain phenomenologically the isotope effect, dome-shaped temperature-dependent dielectric constant near the transition temperature ${T}_{c}$ and the ``crossover'' behavior (first order to second order) in ${\mathrm{H}}_{2}$${\mathrm{C}}_{4}$${\mathrm{O}}_{4}$ crystals. The elongation of the O-H... O bonds alone, contrary to the implication of Samara and Semmingsen [J. Chem. Phys. 71, 1401 (1979)], is not sufficient to explain the isotope effect and other peculiar features of phase transitions in squaric acid. The model parameters, obtained by fitting the experimental electrical susceptibility data, have also been used to calculate heat capacity (C\ifmmode\bar\else\textasciimacron\fi{}=A\ensuremath{\partial}/\ensuremath{\partial}T[T\ensuremath{\chi}(T)]) and reproduces a maximum value of C\ifmmode\bar\else\textasciimacron\fi{} at \ensuremath{\approxeq}7.0 K above ${T}_{c}$ for the ${\mathrm{H}}_{2}$${\mathrm{C}}_{4}$${\mathrm{O}}_{4}$, which agrees with the corresponding experimental value. From our theoretical observation a unified character of the antiferroelectric transition in ${\mathrm{H}}_{2}$${\mathrm{C}}_{4}$${\mathrm{O}}_{4}$ and the antiferromagnetic transitions in some two-dimensional Ising systems is well exposed.