Assume $n\ge3$. Separate the odd and even indices:

$$U(t)=\prod_{i\text{ odd}}(t-x_i),\qquad V(t)=\prod_{i\text{ even}}(t-x_i).$$

Put $P=UV$ and $W=U'V-UV'$. The condition says

$$P'(x_i)=(-1)^{n-i}(x_i^2+1).$$

At an odd-indexed root, $W(x_i)=P'(x_i)$; at an even-indexed root, $W(x_i)=-P'(x_i)$. Hence

$$W(x_i)=(-1)^{n-1}(x_i^2+1)$$

for all $n$ distinct roots.

Both $W$ and $(-1)^{n-1}(t^2+1)$ have degree less than $n$. Their difference has $n$ roots, so

$$\boxed{U'V-UV'=(-1)^{n-1}(t^2+1).}$$