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Ultralight dark matter is a compelling dark matter candidate. In this work, we examine the impact of quadratically-coupled ultralight dark matter on the predictions of Big Bang Nucleosynthesis. The presence of ultralight dark matter can modify the effective values of fundamental constants during Big Bang Nucleosynthesis, modifying the predicted abundances of the primordial elements such as Helium-4. We improve upon the existing literature in two ways: firstly, we take into account the thermal mass acquired by the ultralight dark matter due to its quadratic interactions with the Standard Model bath, which affects the cosmological evolution of the dark matter. Secondly, we treat the weak freeze-out using the full kinetic equations instead of using an instantaneous approximation. Both improvements were shown to impact the Helium-4 prediction in the context of universally-coupled dark matter in previous work. We extend these lessons to more general couplings. We show that with these modifications, Big Bang Nucleosynthesis provides strong constraints of ultralight dark matter with quadratic couplings to the Standard Model for a large range of masses as compared to other probes of this model, such as equivalence principle tests, atomic and nuclear clocks, as well as astrophysical and other cosmological probes.