Deep Equilibrium Models (DEQs) are implicit neural networks with fixed points, which have recently gained attention for learning image regularization functionals, particularly in settings involving Gaussian fidelities, where assumptions on the forward operator ensure contractiveness of standard (proximal) Gradient Descent operators. In this work, we extend the application of DEQs to Poisson inverse problems, where the data fidelity term is more appropriately modeled by the Kullback--Leibler divergence. To this end, we introduce a novel DEQ formulation based on Mirror Descent defined in terms of a tailored non-Euclidean geometry that naturally adapts with the structure of the data term. This enables the learning of neural regularizers within a principled training framework. We derive sufficient conditions and establish refined convergence results based on the Kurdyka--Lojasiewicz framework for subanalytic functions with non-closed domains to guarantee the convergence of the learned reconstruction scheme and propose computational strategies that enable both efficient training and parameter-free inference. Numerical experiments show that our method outperforms traditional model-based approaches and it is comparable to the performance of Bregman Plug-and-Play methods, while mitigating their typical drawbacks, such as time-consuming tuning of hyper-parameters. The code is publicly available at https://github.com/christiandaniele/DEQ-MD.