**Update 2:** The answers below addressed the question in OP's comment. Based the accepted answer it now seems that removing the bottom of edge of rectangles is not needed. In that case, the option `ChartBaseStyle` gives the desired result (there is no need for custom `ChartElementFunction`s):

    Histogram[data, ChartBaseStyle -> EdgeForm[Thickness[.01]]]

[![enter image description here][1]][1]

    Histogram[data, ChartStyle -> "Pastel", 
     ChartElementFunction -> "GlassRectangle", 
     ChartBaseStyle -> EdgeForm[{Opacity[1, Red], Thickness[.01]}]]
[![enter image description here][2]][2]

**Update:** The `ChartElementFunction` in the original post works only for the default setting for the option `BarOrigin`. The new function `ceF2` works for arbitrary values for the `BarOrigin` option.

    ceF2[cedf_: "GlassRectangle", o : OptionsPattern[]][col_: Black, thickness_: 3] := 
     Module[{or = Charting`ChartStyleInformation["BarOrigin"], ll = Tuples[#][[{1, 2, 4, 3}]]},
       ll = RotateRight[ll, Switch[or, Bottom, 0, Top, 2, Left, 3, Right, 1]] ;
       {ChartElementDataFunction[cedf, o][##], 
        col, AbsoluteThickness[thickness], CapForm["Butt"], Line[ll]}] &

Examples:

    Grid[Partition[Histogram[data, ChartStyle -> 1, ImageSize -> 300, 
         PlotLabel -> Style["BarOrigin -> " <> ToString[#], 16, "Panel"], 
         ChartElementFunction -> ceF2["FadingRectangle", "GradientOrigin" -> Top][
           Dynamic[Darker@CurrentValue["Color"]], 5], 
         BarOrigin -> #] & /@ {Bottom, Top, Left, Right}, 2]]
![Mathematica graphics](https://i.sstatic.net/33A6L.png)

**Original post:**

A more flexible chart element function that modifies built-in chart element functions to add thick lines:

    ClearAll[ceF]
    ceF[cedf_: "GlassRectangle", o : OptionsPattern[]][col_: Black, thickness_: 3] := 
      {ChartElementDataFunction[cedf, o][##], 
       col, AbsoluteThickness[thickness], CapForm["Butt"],
       Line[Tuples[#][[{1, 2, 4, 3}]]]} &

Examples:

    SeedRandom[42];
    data = RandomVariate[NormalDistribution[0, 1], 200];

    Histogram[data, ChartStyle -> 1, ChartElementFunction -> ceF[][]]
![Mathematica graphics](https://i.sstatic.net/1cQa1.png)

    Histogram[data, ChartStyle -> 1, 
     ChartElementFunction -> 
      ceF["FadingRectangle"][Dynamic[Darker@Darker@CurrentValue["Color"]], 5]]
![Mathematica graphics](https://i.sstatic.net/jUajy.png)

    Histogram[data, ChartStyle -> 1, 
     ChartElementFunction -> 
      ceF["FadingRectangle", "GradientOrigin"->Top][Dynamic[Darker@CurrentValue["Color"]], 5]]
![Mathematica graphics](https://i.sstatic.net/4HxeR.png)

    Histogram[data, 5, ChartStyle -> {Red, Green, Blue, Orange, Cyan, Purple}, 
     ChartElementFunction -> 
      ceF["FadingRectangle", "GradientOrigin"->Top][Dynamic[Darker@CurrentValue["Color"]], 5]]
![Mathematica graphics](https://i.sstatic.net/aCsHw.png)


  [1]: https://i.sstatic.net/uOu7g.png
  [2]: https://i.sstatic.net/LYxWF.png