How we use the Charity Engine for a search of square numbers whose difference are again squares?

Question

We want find quadruples [w,x,y,z] where the difference of their squares are all again a square: z^2-y^2, z^2-x^2, z^2-w^2, y^2-x^2, y^2-w^2 and x^2-w^2 are all square numbers.

For this we plan to use the Charity Engine to seach for these quadruples in a distributed manner.

For a jump start or (partial) solution on how to approach, I would be very grateful.

What we have so far: We have already a solution based on C++ that searches for such quadruples (see this GitHub Link). And it searched already up to 2^34. However, a distributed search seems to be much more promising in terms of scalability.