In Boffa set theory, there can be numerous automorphisms of the universe, since there can be numerous distinct Quine atoms, singleton sets $a=\{a\}$, which can be permuted, and these permutations extend to automorphisms of the whole universe.

For example, one can build such a model by starting with a model of ZFCU, and then turning the urelements into Quine atoms. You get a model of Boffa set theory this way, and this theory includes ZF-Reg.

But meanwhile, there can be supertransitive models of MacLane set theory or even ZFC-Reg inside the original model, with multiple Quine atoms. Such a model will be intersectional, but admit automorphisms.

One can easily make a model of ZF-Reg with numerous Quine atoms. Simply begin with a model of ZFCU, with numerous urelements, and then turn the urlements into Quine atoms, which are singleton sets $a=\{a\}$, and the result is a model of ZF-Reg. The atoms can be permuted, and these permutations extend to automorphisms of the whole universe.

But meanwhile, there can be supertransitive models of MacLane set theory or even ZFC-Reg inside the original model, with multiple Quine atoms. Such a model will be intersectional, but admit automorphisms.

For example, consider $M=V_{\omega_1}[A]$ in the original universe, with a set $A$ of urelements. In the new model, these turn into Quine atoms, and this model $M$ is supertransitive, hence intersectional, and a model of MacLane set theory (without Reg) and much more. But permutations of $A$ extend to automorphisms of $M$.