One way is with Region computation:

RegionPlot3D[
 RegionIntersection[Cuboid[{ -2. , -2. , -2. } , { 4. , 4. , 4. } ], 
  HalfSpace[Cross[{0, 1, 1}, {1, 0, 1}], {0, 0, 0}]]]
RegionPlot3D[
 RegionDifference[Cuboid[{ -2. , -2. , -2. } , { 4. , 4. , 4. } ], 
  HalfSpace[Cross[{0, 1, 1}, {1, 0, 1}], {0, 0, 0}]]]

Here is a way:

RegionIntersection[Cuboid[{ -2 , -2 , -2 } , { 4 , 4 , 4 } ], 
  InfinitePlane[{{0, 0, 0}, {0, 1, 1}, {1, 0, 1}}]] // InputForm

I find Region computation very frustrating, because there are glaring holes in functionality. For example, this gives useless output:

cubd = Cuboid[{-2, -2, -2}, {4, 4, 4}];
hlfsp = HalfSpace[{-1, -1, 1}, {0, 0, 0}];
RegionIntersection[cubd, hlfsp]

However, if I replace the HalfSpace with a large enough Simplex I get the intersection as a real Polyhedron:

smpl = Simplex[{{24, -10, 14}, {-8, 22, 14}, {-24, -4, -28}, {20, 
     20, -20}}];
RegionWithin[smpl, RegionIntersection[cubd, hlfsp]]
RegionIntersection[cubd, smpl] // Rationalize // Chop // InputForm

It is irritating that I need to Rationalize and Chop the result.

In your first example, the intersection is being represented symbolically. You can get a mesh back using BoundaryDiscretizeRegion:

cubd = Cuboid[{-2, -2, -2}, {4, 4, 4}];
hlfsp = HalfSpace[{-1, -1, 1}, {0, 0, 0}];
BoundaryDiscretizeRegion[RegionIntersection[cubd, hlfsp]]

You can discretize the region, but then you get a fine mesh, almost 2000 vertices.