What can cause a red , large, hard lump under the skin on the outer region of the upper right thigh?

A few things : If a "red skin lump " is tender, has surrounding redness and enlarging, it may be a skin abscess from a bacterial infection. The body can sometimes "wall" these off without any treatment. A doctor performed incision and drainage may cure these +- antibiotics. Its better to see the doctor early rather than wait to see what your body may do.