In this paper, we explore the Hamilton structures in non-equilibrium chemical reactions, which is modeled as a random time-changed Poisson process on countable states. Transition paths between multiple steady states in a chemical reaction is a rare event that can be characterized via the large deviation principle. Compared with the Hamilton principle, we use the Maupertuis principle to compute the transition paths and the associated energy barriers, i.e., the rate function in the large deviation principle. Based on the corresponding stationary Hamilton-Jacobi equation, we select a proper stationary viscosity solution, which in general is not unique, to explicitly compute the energy barriers and the associated optimal control that realizes a transition path. Using one-dimensional example, we characterize the energy barriers for chemical reactions using a geometric quantity in the phase plane. We also compare the reaction barriers with the one in the diffusion approximation and show that the global energy landscape and energy barriers for non-equilibrium chemical reactions are quite different with its diffusion approximation.