The ornamental design for the hair cutting device housing, as shown and described.
FIG. 1 is a top perspective view of the present hair cutting device housing;
FIG. 2 is a top view of same;
FIG. 3 is a right side view of the hair cutting device housing of FIG. 1, both sides being mirror images of each other;
FIG. 4 is a bottom view of same;
FIG. 5 is a front view of the hair cutting device housing of FIG, 1; and,
FIG. 6 is a rear view of same.
The long, dashed broken lines illustrate the cutting element and the button. The remaining short, dashed broken lines around the unshaded elements illustrate portions of the housing. None of the broken lines form part of the claimed design.