The average human hand cannot produce enough grip force to do such a thing. In this study on grip strength, the maximum key grip (holding an object like you would a key) force listed is 42 pounds.
Wood will probably be the softest material a typical modern mahjong tile is made of, from the link in your post. Looking at the compression strength properties of wood, the softest wood listed compresses at 193 pounds per square inch. Keep in mind that hardwoods like maple (456 psi) or oak (778 psi) would actually be used rather than softwoods.
Plastics take things to an entirely new level. This chart shows the compressive properties of some common polymer plastics. Note that the values are given in MPa, or megapascals. 1 megapascal = 145.037738 pound-force/square inch (PSI).
As for materials such as pulverized bone, the only references I could find involved using the materials as partial replacement for concrete. I'd say it's safe to infer that the compressive strength for the compound used in making mahjong tiles would be at least as strong as hardwoods.
Keep in mind these figures only talk about the compression strength of the material. Just compressing the tile alone would not be enough to remove the engraving and paint of the characters on the tile face. Some shear force would also need to be applied as well. I don't think it's necessary to go there at this point, though.