We claim that the only solutions are $(2,3,3)$ and its permutations.
Factoring the above squares and canceling the terms gives you:
$8(xyz)^2 + 2(x^2 +y^2 + z^2) = 4((xy)^2 + (yz)^2 + (zx)^2) + 2024$
Jumping on the coefficients in front of the $x^2$ , $y^2$ , $z^2$ terms, we factor into:
$(2x^2 - 1)(2y^2 - 1)(2z^2 - 1) = 2023$
Realizing that the only factors of 2023 that could be expressed as $(2x^2 - 1)$ are $1$ , $7$ , and $17$ , we simply find that the only solutions are $(2,3,3)$ by inspection.
-Max
Alternatively, a more obvious factorization is:
$2(x+y+z+2xyz)^2=(2xy+2yz+2zx+1)^2+2023$
$(\sqrt{2}x+\sqrt{2}y+\sqrt{2}z+2\sqrt{2}xyz)^2-(2xy+2yz+2zx+1)^2=2023$
$(2\sqrt{2}xyz+2xy+2yz+2zx+\sqrt{2}x+\sqrt{2}y+\sqrt{2}z+1)(2\sqrt{2}xyz-2xy-2yz-2zx+\sqrt{2}x+\sqrt{2}y+\sqrt{2}z-1)=2023$
$(\sqrt{2}x+1)(\sqrt{2}y+1)(\sqrt{2}z+1)(\sqrt{2}x-1)(\sqrt{2}y-1)(\sqrt{2}z-1)=2023$
$(2x^2-1)(2y^2-1)(2z^2-1)=2023$
Proceed as above. ~eevee9406