We extend our previous analysis of the decays ${B}_{d,s}\ensuremath{\rightarrow}{\ensuremath{\mu}}^{+}{\ensuremath{\mu}}^{\ensuremath{-}}$ in the minimal supersymmetric standard model (MSSM) to include gluino and neutralino contributions. We provide analytic formulas, valid at large values of $\mathrm{tan}\ensuremath{\beta},$ for the scalar and pseudoscalar Wilson coefficients arising from neutral Higgs boson exchange diagrams with gluinos and neutralinos. Together with the remaining contributions ${(W}^{\ifmmode\pm\else\textpm\fi{}},$ ${H}^{\ifmmode\pm\else\textpm\fi{}},$ ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\chi}}}^{\ifmmode\pm\else\textpm\fi{}}),$ and assuming the Cabibbo-Kobayashi-Maskawa (CKM) matrix to be the only source of flavor violation, we assess their implications for the branching fractions $\mathcal{B}({B}_{d,s}\ensuremath{\rightarrow}{\ensuremath{\mu}}^{+}{\ensuremath{\mu}}^{\ensuremath{-}}).$ Of particular interest is the quantity $R\ensuremath{\equiv}\mathcal{B}({B}_{d}\ensuremath{\rightarrow}{\ensuremath{\mu}}^{+}{\ensuremath{\mu}}^{\ensuremath{-}})/\mathcal{B}({B}_{s}\ensuremath{\rightarrow}{\ensuremath{\mu}}^{+}{\ensuremath{\mu}}^{\ensuremath{-}}),$ since (i) the theoretical errors cancel to a large extent, and (ii) it offers a theoretically clean way of extracting the ratio $|{V}_{\mathrm{td}}{/V}_{\mathrm{ts}}|$ in the standard model, which predicts ${R}_{\mathrm{SM}}\ensuremath{\sim}|{V}_{\mathrm{td}}{/V}_{\mathrm{ts}}{|}^{2}\ensuremath{\sim}{O(10}^{\ensuremath{-}2}).$ Exploring three different scenarios of modified minimal flavor violation $(\overline{\mathrm{MFV}}),$ we find that part of the MSSM parameter space can accommodate large ${B}_{d,s}\ensuremath{\rightarrow}{\ensuremath{\mu}}^{+}{\ensuremath{\mu}}^{\ensuremath{-}}$ branching fractions while being consistent with various experimental constraints. More importantly, we show that the ratio R can be as large as $O(1),$ while the individual branching fractions may be amenable to detection by ongoing experiments. We conclude that within the MSSM with large $\mathrm{tan}\ensuremath{\beta}$ the decay rates of ${B}_{d}\ensuremath{\rightarrow}{\ensuremath{\mu}}^{+}{\ensuremath{\mu}}^{\ensuremath{-}}$ and ${B}_{s}\ensuremath{\rightarrow}{\ensuremath{\mu}}^{+}{\ensuremath{\mu}}^{\ensuremath{-}}$ can be of comparable size even in the case where flavor violation is due solely to the CKM matrix.