We investigate the combinatorial and geometric properties of automorphism groups of universal right-angled Coxeter groups, which are the automorphism groups of free products of copies of Z_2. It is currently an open question as to whether or not these automorphism groups have non-positive curvature. Analogous to Outer Space as a model for Out(F_n), we prove that the natural combinatorial and topological model for their outer automorphism groups can not be given an equivariant CAT(0) metric.  This is particularly interesting as there are very few non-trivial examples of proving that a model space of independent interest is not CAT(0).