**Update 2:**  Based on the accepted answer it now seems that removing the bottom of edge of rectangles is not essential. 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