In this paper, we continue our analysis of the chaotic four-body problem by presenting a general ansatz-based analytic treatment using statistical mechanics, where each outcome of the four-body problem is regarded as some variation of the three-body problem (e.g., when two single stars are produced, called the 2+1+1 outcome, each ejection event is modeled as its own three-body interaction by assuming that the ejections are well separated in time). This is a generalization of the statistical mechanics treatment of the three-body problem based on the density-of-states formalism. In our case, we focus on the interaction of two binary systems, after which we divide our results into three possible outcome scenarios (2+2, 2+1+1, and 3+1). For each outcome, we apply an ansatz-based approach to deriving analytic distribution functions that describe the properties of the products of chaotic four-body interactions involving point particles. To test our theoretical distributions, we perform a set of scattering simulations in the equal-mass point particle limit using FEWBODY. We compare our final theoretical distributions to the simulations for each particular scenario, finding consistently good agreement between the two. The highlights of our results include that binary-binary scatterings act to systematically destroy binaries producing instead a single binary and two ejected stars or a stable triple, the 2+2 outcome produces the widest binaries and the 2+1+1 outcome produces the most compact binaries.