We study $SU(3)$-breaking effects in the neutral ${B}_{d}$-${\overline{B}}_{d}$ and ${B}_{s}$-${\overline{B}}_{s}$ systems with unquenched ${N}_{f}=2+1$ lattice quantum chromodynamics (QCD). We calculate the relevant matrix elements on the MILC collaboration's gauge configurations with asqtad-improved staggered sea quarks. For the valence light-quarks ($u$, $d$, and $s$) we use the asqtad action, while for $b$ quarks we use the Fermilab action. We obtain $\ensuremath{\xi}={f}_{{B}_{s}}\sqrt{{B}_{{B}_{s}}}/{f}_{{B}_{d}}\sqrt{{B}_{{B}_{d}}}=1.268\ifmmode\pm\else\textpm\fi{}0.063$. We also present results for the ratio of bag parameters ${B}_{{B}_{s}}/{B}_{{B}_{d}}$ and the ratio of Cabibbo-Kobayashi-Maskawa matrix elements $|{V}_{td}|/|{V}_{ts}|$. Although we focus on the calculation of $\ensuremath{\xi}$, the strategy and techniques described here will be employed in future extended studies of the $B$ mixing parameters $\ensuremath{\Delta}{M}_{d,s}$ and $\ensuremath{\Delta}{\ensuremath{\Gamma}}_{d,s}$ in the standard model and beyond.