Rao-Blackwell's Theorem for uniform distribution

Here the first step is to choose an arbitrary $X_i$.

Note that

\begin{equation*} \begin{split} & \hspace{15pt} E(X_i \ | \ X_{(1)} = l, \ X_{(n)} = u) \\ & = E(X_i \ |\ X_{(1)} = l, \ X_{(n)} = u, \ X_{i} = l) * P(X_i = X_{(1)}) \\ & + \ E(X_i \ | \ X_{(1)} = l, \ X_{(n)} = u, \ X_i = u) * P(X_i = X_{(n)}) \\ & + \ E(X_i \ | \ X_{(1)} = l, \ X_{(n)} = u, \ l < X_i < u) * P (X_i \neq X_{(1)}, X_i \neq X_{(n)}) \\ & = l*\frac{1}{n}+u*\frac{1}{n}+\frac{l+u}{2}*\frac{n-2}{n} \\ &= \frac{(l + u)}{2} \end{split} \end{equation*}

Now just apply the linearity of expectation to get that $E(\bar{X} \ | \ X_{(1)}, X_{(n)}) = \frac{X_{(1)} + X_{(n)}}{2}$

Edit: tweaked to make it clear that we are consider the probability $X_i$ is equal to the order statistic.