Closed form solution for the zeros of an infinite sum
Solution 1:
If you decompose $\frac{1}{n^4 - x^2}$ into $\frac{1}{2x}(\frac{1}{n^2 - x} - \frac{1}{n^2 + x})$, and compute both series independently (everything converges absolutely, so it is OK to do so), you'd end up with an equation
$$\pi\sqrt{x}(\coth{\pi\sqrt{x}} + \cot{\pi\sqrt{x}}) = 2$$
which doesn't look promising for a closed form. It might be a good starting point for a numerical solution.