Since the left hand side $2x^2y^2 +2y^2z^2 +2z^2x^2 -x^4 -y^4 -z^4$ is $(-x+y+z)(x+y-z)(x-y+z)(x+y+z)$ and since the right hand side is $576=2^6\cdot3^2$, this boils down to case checking. Since the four factors $(-x+y+z)$, $(x+y-z)$, $(x-y+z)$, $(x+y+z)$ all have the same parity, they must all be even. Furthermore, the three smaller factors $(-x+y+z)$, $(x+y-z)$, $(x-y+z)$ add up to the largest factor $(x+y+z)$, this only leaves the four factors $2\cdot1=2$, and $2\cdot2=4$, and $2\cdot3=6$, and $2\cdot6=12$. The positive integer solutions hence are $(x,y,z)=(3,4,5)$ together with all permutations.