Are there any nontrivial integer solutions $c,d$ for which $$f(c,d)=|-c^4 - 16 c^3 d + 6 c^2 d^2 + 16 c d^3 - d^4|$$ and $$g(c,d)=|-c^4 + c^3 d + 6 c^2 d^2 - c d^3 - d^4|$$ are simultaneously squares of integers?
By nontrivial, I mean cases where $c,d$ are nonzero and coprime. To avoid any confusion, I explicitly rule out $(c,d) = (1,1),(1,-1),(-1,1),(-1,-1)$.
My bet is no. Here is Mathematica code checking that all pairs of coprime integers of magnitude less than $1000$ do not solve the equation.
CoprimeSet[n_, m_] :=
Pick[#, CoprimeQ @@@ #] &[Subsets[Range[n], {m}]];
mset = CoprimeSet[1000, 2];
mset2 = Transpose[{1, -1}*Transpose[mset]];
mtemX[c_, d_] := -c^4 - 16 c^3 d + 6 c^2 d^2 + 16 c d^3 - d^4;
mtemY[c_, d_] := - c^4 + c^3 d + 6 c^2 d^2 - c d^3 - d^4;
Total[Boole[Sqrt[Abs[mtemX[#[[1]], #[[2]]]] ] \[Element] Integers]*
Boole[Sqrt[Abs[mtemY[#[[1]], #[[2]]]]] \[Element] Integers] & /@
mset]
Total[Boole[Sqrt[Abs[mtemX[#[[1]], #[[2]]]] ] \[Element] Integers]*
Boole[Sqrt[Abs[mtemY[#[[1]], #[[2]]]]] \[Element] Integers] & /@
mset2]
While these expressions might seem like they come from nowhere, they're actually motivated by a recent MO question that in part is asking whether $x^4 + y^4 = 17 z^4$ has nontrivial positive integer solutions besides $(x,y,z)=(1,2,1)$. If both $f(c,d)$ and $g(c,d)$ can be simultaneously square, one can construct another nontrivial solution to $x^4 + y^4 = 17 z^4$. I suspect there won't be an easy answer to my question above, but if there is, that could help answer that MO question.
Indeed, there are no nonzero, coprime choices for $c,d$ for which $f(c,d)$ and $g(c,d)$ are both squares of integers. This follows from the discussion in the question and Flynn and Wetherell's proof that all of the integer solutions to $x^4 + y^4 = 17 z^4$ are trivially related to that of $(x,y,z) = (2,1,1)$.
However, this might not feel entirely satisfactory, so I will leave this question open for answers that more directly tackle the problem.