ClearAll[cF]
cF[cedf_: "BoxWhisker", o : OptionsPattern[]] :=
Dynamic@With[{col = Darker@Darker@CurrentValue["Color"]}, {EdgeForm[{col, Thick}],
ChartElementDataFunction[cedf, o][##],
ListPlot[Thread[{Mean[#[[1]]], #2}], PlotStyle -> col, PlotMarkers -> #3[[1]]][[1]]}] &
Examples:
SeedRandom[3]
dt = RandomVariate[NormalDistribution[], {5, 10}];
markers = Graphics`PlotMarkers[][[;; 5]];
BoxWhiskerChart[Thread[dt -> markers], {{"MeanMarker", 1}, {"MedianMarker", None}},
ChartStyle -> 1, ChartElementFunction -> cF[]]
BoxWhiskerChart[Thread[dt -> markers], {{"MeanMarker", 1}, {"MedianMarker", None}},
ChartStyle -> 1, ChartBaseStyle -> FaceForm[],
ChartElementFunction -> cF[]]
