In this paper, we explore the hull of linear codes over the non-unitary rings of order four, namely $\mathit{E}$, $\mathit{I}$ and $\mathit{H}$. Initially, we determine the binary associated codes of the hull over each ring; then, we characterize the hulls in terms of these codes. Furthermore, we establish the connection between the hull of any linear code over these rings and the hull of its dual. Finally, we classify linear codes over each ring with prescribed hull sizes under permutation equivalence for short lengths.