Let's start with the maximum frame and use the frame operator as defined in the other thread. Suppose we can prove that, based on this definition, the frames of the product of balls of any dimension are given by a Pascal's triangle kind of pattern. e.g.
BxB
SxB U BxS
SxS
BxBxB
SxBxB U BxSxB U BxBxS
SxSxB U SxBxS U BxSxS
SxSxS
etc.
Then the easiest way to work out the homology groups of all frames would be to come up with some analogue of Mayer-Vietoris that uses unions of three or more shapes. I'll ask my lecturer if this is possible when I see him. In the meantime I can work out products of two balls.
The homology groups of X = (a+1)x(b+1) are
H X = h0
H fX = h0+ha+b+1 EDIT: fixed this equation
H f2X = h0+ha+hb+ha+b
This works for any a and b, even if a=0 or a=b.