Step 1: Use the square midpoint fact.
For a square $ABCD$ the diagonals $AC$ and $BD$ share the same midpoint. We will use this to find $\lambda$ and $\mu$.
Step 2: Write the midpoints.
With $A=(0,\lambda,1)$, $B=(\mu,3,-1)$, $C=(\lambda,5,0)$, $D=(\mu,6,\mu)$, the midpoint of $AC$ is $\left(\tfrac{\lambda}{2},\tfrac{\lambda+5}{2},\tfrac12\right)$ and of $BD$ is $\left(\mu,\tfrac{9}{2},\tfrac{\mu-1}{2}\right)$.
Step 3: Match the y-parts.
\[ \frac{\lambda+5}{2}=\frac{9}{2}\ \Rightarrow\ \lambda=4. \]
Step 4: Match the x-parts.
\[ \mu=\frac{\lambda}{2}=2. \]
Step 5: Check the z-parts.
$\tfrac{\mu-1}{2}=\tfrac{2-1}{2}=\tfrac12$, which matches the AC midpoint z-value. Good, so $\lambda=4,\ \mu=2$.
Step 6: Find a side length.
Side $AB$ with $A(0,4,1)$ and $B(2,3,-1)$:
\[ AB=\sqrt{2^2+(-1)^2+(-2)^2}=\sqrt{9}=3. \]
\[ \boxed{3} \]