I consider this more a "recreational math" problem, possibly lacking a solution (because it stems from the question of "magic square of squares") or simply intractable with reasonable effort.
Consider the following matrix-equation in integers with otherwise arbitrary choosable values for $e,h,i$ only with the additional condition, that they must all be different, all must be odd, and more precisely, $e^2,h^2,i^2$ must be $1 \pmod {12}$ $$ \begin{array} {rcrrrr} \begin{bmatrix} a^2\\b^2\\c^2\\d^2\\f^2\\g^2 \end{bmatrix} & =& \left[ \begin{array} {rrr} 2 & 0 & -1\\ 2 & -1 & 0 \\ -1 & 1 & 1 \\ -2 & 1 & 2 \\ 4 & -1 &-2 \\ 3 & -1 & -1 \\ \end{array} \right] & \cdot \begin{bmatrix} e^2\\ h^2 \\ i^2 \end{bmatrix} \end{array} $$ Is it possible to find some $e,h,i$ with the given conditions, such that the lhs contains only squares? I thought, it might by possible by analysis of the coefficients matrix, but after some fiddling I didn't see a promising path... On the other hand, there might be some obvious(?) argument, that such a combination of squares is impossible. Or - that the given coefficients-matrix cannot -for some good reason- be of help for this problem.
[update] One could extend the conditions, such that we describe the unknowns $d^2,f^2,g^2$ as dependent on $a^2,b^2,c^2$ by the following further matrix equation $$ \begin{array} {} 9 \cdot \begin{bmatrix} d^2\\f^2\\g^2\end{bmatrix} &=& \left[ \begin{array} {rrr} -6 & 3 & 12 \\ 12 & 3 & -6 \\ 6 & 6 & -3 \end{array} \right] \cdot \begin{bmatrix} a^2 \\ b^2 \\ c^2 \end{bmatrix} \end{array} $$