With some effort, Mathematica evaluates this as
$$\int_{0}^{1} \int_{0}^{1} \frac{(\log(1+x^2)-\log(1+y^2))^2 }{|x-y|^{2}}dx dy=\frac{1}{192} \left(384 \sqrt{2} \, _4F_3\left(\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2};\frac{3}{2},\frac{3}{2},\frac{3}{2};\frac{1}{2}\right)-2 \left(\log \left(2^{96 C+\pi ^2 (8+21 i)}\right)+192 i \text{Li}_3\left(\frac{1}{2}+\frac{i}{2}\right)+\log \left(2^{-24 \left(\pi \log \left(\sqrt{2}+2\right)-8 i \text{Li}_2\left(-\sqrt[4]{-1}\right)\right)}\right)+(828-105 i) \zeta (3)\right)+12 \pi \left(64 C-3 i-7 \log ^2(2)\right)+8 i \left(48 \text{Li}_2\left(1-\sqrt[4]{-1}\right) \log (2)+\log ^3(2)\right)-23 \pi ^3\right)=0.572532$$
I'll see if I can massage this to a slightly more pleasant expression.