The authors study the nonlinear initial-boundary value problem \(u_t=\Delta u - u^{-q},\) \(u(x,0)=u_0(x)\) on the bounded domain \(\Omega \subset {\mathbb R}^n\) with a smooth boundary and the Neumann boundary condition \(\partial u / \partial \nu=u^p,\) where \(p\) and \(q\) are positive parameters, \(0\leq u_0(x) \leq M,\) and \(\partial u_0 / \partial \nu =u_0^p\) on \(\partial \Omega.\) They derive criteria for finite time quenching (i.e. convergence of the solution to zero) and blow-up, estimate the rates of quenching and blow-up, and determine the sets of points in which quenching or blow-up can occur. The results parallel earlier analyses of this system with zero flux or zero reaction part.