Estimating variance from a single pair of measurements

You take just two independent readings $X_1, X_2$ from a process with unknown mean and variance $\sigma^2$. A colleague proposes estimating the variance with

$\hat\sigma^2 = \tfrac{1}{2}(X_1 - X_2)^2.$

Is this an unbiased estimator of $\sigma^2$? Prove it, and connect it to the usual "divide by $n-1$" rule.