A work-around: add the vertical lines using `Graohics3D`:


    Show[ListPointPlot3D[{xyzData, predictedZ}, PlotRange -> All, 
      PlotStyle -> {Directive[Black, PointSize[Medium]], 
        Directive[Red, PointSize[Small]]}], 
     Plot3D[model[x, y], {x, xMin, xMax}, {y, yMin, yMax}, 
      PlotStyle -> Directive[Opacity[0.1], Specularity[], Glow[White]], 
      Mesh -> 8, ViewVertical -> {0, 0, 1}], 
     Graphics3D[{Blue, Thick, Line /@ Transpose[{xyzData, predictedZ}]}], 
     PlotRange -> All, AxesLabel -> {"X", "Y", "Z"}, 
     LabelStyle -> Directive[FontFamily -> "Helvetica"]]

![Mathematica graphics](https://i.sstatic.net/zu3ES.png)