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[]]

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[]]