The eigenfunctions are sine waves of the right wavelength. The energy operator is essentially a second derivative. The directions x,y,z act independently, so you will have a complete eigenfunction that's a product of an X part, a Y part and a Z part. To see the degeneracy, just find out how many ways there are to write a particular energy. For example:
Energy
1 = 1^2+0+0 (x contribution =1, y=0,z=0)
1 = 0 + 1 + 0
1 = 0 + 0 + 1, so degeneracy = 3 here.
For higher values, there may be other ways to write the energy, for instance:
2 = 1 + 1 + 0 ( & permutations)
3 = 1 + 1 + 1 (only 1 way )
etc.
