  The two dimensional square lattice hard-core boson Hubbard model with near
neighbor interactions has a `checkerboard' charge density wave insulating phase
at half-filling and sufficiently large intersite repulsion. When doped, rather
than forming a supersolid phase in which long range charge density wave
correlations coexist with a condensation of superfluid defects, the system
instead phase separates. However, it is known that there are other lattice
geometries and interaction patterns for which such coexistence takes place. In
this paper we explore the possibility that anisotropic hopping or anisotropic
near neighbor repulsion might similarly stabilize the square lattice
supersolid. By considering the charge density wave structure factor and
superfluid density for different ratios of interaction strength and
hybridization in the $\hat x$ and $\hat y$ directions, we conclude that phase
separation still occurs.
