We consider diffusion in a potential well with a boundary that randomly switches between absorbing and reflecting and show how the switching boundary affects the classical escape theory. Using the theory of stochastic hybrid systems, we derive boundary value problems for the mean first passage time and splitting probability and find explicit solutions in terms of the spectral decomposition of the associated differential operator. Further, using a more probabilistic approach, we prove asymptotic formulae for these statistics in the small diffusion limit. In particular, we show that the statistical behavior depends critically on the gradient of the potential near the switching boundary and we derive corrections to Kramers' reaction rate theory.