We investigate the freezing behavior of particles interacting with a Yukawa potential using extensions of the Denton and Ashcroft modified weighted density approximation (MWDA) model of density functional theory [A. R. Denton and N. W. Ashcroft, Phys. Rev. A 39, 470 (1989)]. An attempt is made to incorporate properties of the static solid into the fluid-based MWDA model via our previous model for the crystallization of inverse nth-power fluids [D. C. Wang and A. P. Gast, J. Chem. Phys. 110, 2522 (1999)], as well as a model that includes the Einstein vibrations of the localized particles. Both extensions yield improvements over the MWDA model in terms of coexisting densities and the ability to stabilize a body-centered cubic solid compared with computer simulation data. The fractional change in density upon freezing also compares favorably with results from available simulation studies and those for the inverse nth-power system. Reasons for the differences in results obtained for freezing properties of the Yukawa system among computer simulation data, theoretical approaches, and experimental studies are discussed.