In a recent study we have reported a new type of trial wavefunction symmetric under the exchange of particles, which is able to describe a supersolid phase. In this work, we use the diffusion Monte Carlo method and this model wavefunction to study the properties of solid (4)He in two- and quasi-two-dimensional geometries. In the purely two-dimensional (2D) case, we obtain results for the total ground-state energy and freezing and melting densities which are in good agreement with previous exact Monte Carlo calculations performed with a slightly different interatomic potential model. We calculate the value of the zero-temperature superfluid fraction ρ(s)/ρ of 2D solid (4)He and find that it is negligible in all the considered cases, similarly to what is obtained in the perfect (free of defects) three-dimensional crystal using the same computational approach. Interestingly, by allowing the atoms to move locally in the direction perpendicular to the plane where they are confined to zero-point oscillations (quasi-2D crystal), we observe the emergence of a finite superfluid density that coexists with the periodicity of the system.