  In this paper, we analyze a real-valued reflected backward stochastic
differential equation (RBSDE) with an unbounded obstacle and an unbounded
terminal condition when its generator $f$ has quadratic growth in the
$z$-variable. In particular, we obtain existence, comparison, and stability
results, and consider the optimal stopping for quadratic $g$-evaluations. As an
application of our results we analyze the obstacle problem for semi-linear
parabolic PDEs in which the non-linearity appears as the square of the
gradient. Finally, we prove a comparison theorem for these obstacle problems
when the generator is convex or concave in the $z$-variable.
