So for reference, I have no previous experience with Diophantine equations, but for this particular set I believe I have found something that works.I also didn't exactly set out to solve this system, more stumbled across it. This Diophantine equation is essentially this:
$$A^2+B^2 =C^2\\ A^2+D^2 =E^2 \\ B^2+D^2 =F^2\\ B^2+E^2 =G^2.$$
So, my mind should have first went to the fact that these all look like the pythagorean equation, but it didn't so I solved them all for zero and created an augmented matrix to try and get some more information. This told me that
$G^2, F^2$, and $E^2$ are all free choices, $A^2=G^2-F^2, B^2=G^2-E^2, D^2=-G^2+F^2+E^2$, and $C^2=2G^2-F^2-E^2$. This is similarly awkward, but I figured as long as there are 3 squares that add up to each other they should be able to get a solution. So, I tried setting $G=5, F=4$, and $E=3$ and it worked. But these aren't the only set of numbers that would work, they could be any Pythagorean triple $(x, y, z)$ and as long as $G=x, F=y, E=z$ and $x$ is the hypotenuse. From this, I found that $A=z, B=y, C=x$, and $D=0$. This works when plugged into the original set of equations.
$$z^2+y^2 =x^2 \\ z^2+0 =z^2 \\ y^2+0 =y^2 \\ y^2+z^2 =x^2$$
These are only the set of positive integers that are solutions, I'm sure there are many more but like I said I more stumbled across this. Is this correct, or am I spewing nonsense?