Step-by-step Solution:

(1) To verify the square difference between the largest and smallest number in three consecutive integers is a multiple of $4$, let's calculate $5^{2}-3^{2}$:

$$\begin{align*} 5^{2}-3^{2} & = 25-9 \\ & = 16 \\ & = 4 \times 4. \end{align*}$$

Therefore, the result, $16$, is indeed a multiple of $4$. Thus, the statement is verified for this specific case, and the answer is $\boxed{4}$.

(2) To prove that in any three consecutive odd numbers, the square difference between the largest and smallest number is a multiple of $8$, let's denote the three consecutive odd numbers as $2n-1$, $2n+1$, and $2n+3$:

$$\begin{align*} \left(2n+3\right)^{2}-\left(2n-1\right)^{2} & = \left(2n+3+2n-1\right)\left(2n+3-2n+1\right) \\ & = \left(4n+2\right)\left(4\right) \\ & = 8\left(2n+1\right). \end{align*}$$

This calculation shows that the square difference between the largest and smallest number in any three consecutive odd numbers is indeed $8\left(2n+1\right)$, which is a multiple of $8$. Therefore, the proof is complete, and we can conclude that the square difference between the largest and smallest number in any three consecutive odd numbers is a multiple of $\boxed{8}$.