Reflection colors are lost on high resolution MeshRegion

I think that the option PlotTheme for large MeshRegions is automatically set to "LargeMesh" for performance reasons. Try the following:

region00 = 
 DiscretizeRegion[
  Polygon[{{-1, -1, 0}, {1, -1, 0}, {1, 1, 0}, {-1, 1, 0}}], 
  MaxCellMeasure -> 0.000003, 
  MeshCellStyle -> {1 -> Black, 2 -> Directive[Specularity[Yellow, 100], Blue]},
  PlotTheme -> "SmoothShading"]  

One can also set the system option for "LargeMeshThreshold" to be greater than the number of mesh cells (default threshold is 1000):

SetSystemOptions["RegionOptions" -> {"LargeMeshThreshold" -> 2000}];
Show[region00, lightSource, Axes -> True, 
 AxesLabel -> {"X", "Y", "Z"}, 
 Lighting -> {{"Point", White, lightSourcePosition}}, 
 ImageSize -> 600, ViewCenter -> {0.5`, 0.5`, 0.5`}, 
 ViewPoint -> {0.027, -3.14, 1.2}, 
 ViewVertical -> {-0.017, -0.49, 14.5}]