We prove by induction on $n$.

When $n=3$, we can assume that $x_{1}>x_{2}>x_{3}$ (only when $n=3$) and the LHS $\geq 1^{2}+1^{2}+2^{2}=6$.

Moving from $n$ to $n+1$, the change in LHS is
$$C_{n+1}=\left(x_{n}-x_{n+1}\right)^{2}+\left(x_{n+1}-x_{1}\right)^{2}-\left(x_{n}-x_{1}\right)^{2} .$$
We need to show that $C_{n+1} \geq 4$. We can see that it is not true in general. However, due to the cyclicity, we can assume that $x_{n+1}$ is the least number.

Moreover, since the LHS is unchanged under translation, we can assume also that $x_{n+1}=0$. Thus, we have
$$C_{n+1}=x_{n}^{2}+x_{1}^{2}-\left(x_{n}-x_{1}\right)^{2}=2 x_{1} x_{n} \geq 4 .$$