  Recent advances in $b$, $c$, and $s$ quark tagging coupled with novel
statistical analysis techniques will allow future high energy and high
statistics electron-positron colliders, such as the FCC-ee, to place
phenomenologically relevant bounds on flavor violating Higgs and $Z$ decays to
quarks. We assess the FCC-ee reach for $Z/h\to bs, cu$ decays as a function of
jet tagging performance. We also update the SM predictions for the
corresponding branching ratios, as well as the indirect constraints on the
flavor violating Higgs and $Z$ couplings to quarks. Using type III two Higgs
doublet model as an example of beyond the standard model physics, we show that
the searches for $h\to bs, cu$ decays at FCC-ee can probe new parameter space
not excluded by indirect searches. We also reinterpret the FCC-ee reach for
$Z\to bs , cu$ in terms of the constraints on models with vectorlike quarks.
