We consider the Dirichlet problem for a class of fully nonlinear elliptic equations on a bounded domain $\Omega$. We assume that $\Omega$ is symmetric about a hyperplane $H$ and convex in the direction perpendicular to $H$. By a well-known result of Gidas, Ni and Nirenberg and its generalizations,  all positive solutions are reflectionally symmetric about $H$ and decreasing  away from the hyperplane in the direction orthogonal $H$. For nonnegative solutions, this result is not always true.  We show that, nonetheless, the symmetry part of the result remains valid for nonnegative solutions: any nonnegative solution $u$ is symmetric about $H$. Moreover, we prove that if $u\not\equiv 0$, then the nodal set of $u$ divides the domain $\Omega$ into a finite number of reflectionally symmetric subdomains in which  $u$  has the usual Gidas-Ni-Nirenberg symmetry and monotonicity properties. We also show several examples of  nonnegative solutions with a nonempty interior nodal set.