We prove that the $\varphi^4_3$ model satisfies a version of Segal's axioms in the special case of three-dimensional tori and cylinders. As a consequence, we give the first proof that this model satisfies a Markov property and we characterize its boundary law up to absolutely continuous perturbations. In addition, we use Segal's axioms to give an alternative construction of the $\varphi^4_3$ Hamiltonian on two-dimensional tori as compared with Glimm (Comm. Math. Phys., 1968). We exploit this probabilistic approach to prove novel fundamental spectral properties of the Hamiltonian, such as discrete spectrum and a Perron-Froebenius type result on its ground state. The key technical contributions of this article are the development of tools to analyze $\varphi^4_3$ models with rough boundary conditions. We heavily use the variational approach to $\varphi^4_3$ models introduced in Barashkov and Gubinelli (Duke, 2020) that is based on the Boué-Dupuis formula and dual to Polchinski's continuous renormalization group.