  When hexagonal boron nitride (hBN) and graphene are aligned at zero or small
twist angle, a moir\'e structure is formed due to the small lattice constant
mismatch between the two structures. In this work, we analyze magnetic ordering
tendencies, driven by onsite Coulomb interactions, of encapsulated bilayer
graphene (BG) forming a moir\'e structure with one (hBN-BG) or both hBN layers
(hBN-BG-hBN), using the random phase approximation. The calculations are
performed in a fully atomistic Hubbard model that takes into account all
$\pi$-electrons of the carbon atoms in one moir\'e unit cell. We analyze the
charge neutral case and find that the dominant magnetic ordering instability is
uniformly antiferromagnetic. Furthermore, at low temperatures, the critical
Hubbard interaction $U_c$ required to induce magnetic order is slightly larger
in those systems where the moir\'e structure has caused a band gap opening in
the non-interacting picture, although the difference is less than 6%.
Mean-field calculations are employed to estimate how such an
interaction-induced magnetic order may change the observable single-particle
gap sizes.
