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I am coloring several countries in the manner of the following:

myc[name_] := If[name == "UnitedStates", Red, Lighter[Gray]] 
Graphics[{myc[#], CountryData[#, {"FullPolygon", "WinkelTripel"}]} & /@ CountryData[]] 

How can I add country borders and longitude-latitude grid, as we could do with WorldPlot?

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Specifying EdgeForm resolves the country borders. Latitude and longitude...well here is a clumsy way. Depending on your desired grid you may have to clean up conversion.

myc[name_] := If[name == "UnitedStates", Red, Lighter[Gray]]
lat = Quiet[
   Line /@ Table[
     Table[GeoGridPosition[GeoPosition[{j, k, 0}], 
        "WinkelTripel"][[1, {1, 2}]], {j, -90, 90, 5}], {k, -170, 170,
       15}]];
long = Quiet[
   Line /@ Table[
     Table[GeoGridPosition[GeoPosition[{k, j, 0}], 
        "WinkelTripel"][[1, {1, 2}]], {j, -170, 170, 15}], {k, -90, 
      90, 10}]];
Graphics[{Gray, lat, long, myc[#], EdgeForm[Black], 
    CountryData[#, {"FullPolygon", "WinkelTripel"}]} & /@ 
  CountryData[]]

enter image description here

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