So if I have a circle in 3D space, of which I know just 4 points on the edge of the circle that are 90 degrees from each other. I want to know the maximum distance between this circle in 3D and a plane that lies directly below it on the same axis.
I'm not sure where to start I was thinking to try and find the equation of the circle but I don't know how to do that in 3D. Once we have that how do you find the max distance from any point in the circle to the plane below.
Let's call your $4$ points $X_1,X_2,X_3,X_4$, and the plane $\mathcal P$.
In your configuration, it is fairly easy to find the center and radius of your circle. Your four points indeed define two diameter of the circle. Let $C$ be the center.
Now that you know the center, you can easily find two orthogonal vectors that are parallel to the plane that contains the circle. Just pick an arbitrary point, say $X_1$, and some other point $X_i$ such that $X_1X_j$ is not a diameter. Then the two vectors $\vec{CX_1}$ and $\vec{CX_i}$ are orthogonal, with norm equal to the radius of the circle.
From there, a point $M$ is on the circle if and only if there exists a real number $\theta\in\mathbb R$ such that $$\vec{CM} = \cos(\theta) \vec{CX_1} + \sin(\theta) \vec{CX_i}$$
This gives you a parametric equation of the circle, from which you can probably derive the point on the circle with maximum distance to $\mathcal P$.
I personally don't really like the solution I suggested above, but I think it fits your initial attempt/idea better than what I had in mind initially.