I'm still trying to get some feeling about this question...
Given Jesse Peterson's answer to this question (he showed that $\phi\circ\phi\sim\phi$ is impossible), I suspect that the following is also impossible. But I'm unable to generalize his argument.
Let $M$ be a type III factor, and let $\phi:M\to M$ be an irreducible endomorphism (the relative commutant of $\phi(M)$ in $M$ is trivial).
Let $v_1$, $v_2\in M$ be isometries with orthogonal ranges summing up to $1$
($v_1^*v_1=v_2^*v_2=v_1v_1^*+v_2v_2^*=1$). Define $$\phi\oplus\phi:m\mapsto v_1\phi(m)v_1^*+v_2\phi(m)v_2^*.$$
Question: Is it possible to have $\phi\circ\phi$ conjugate to $\phi\oplus\phi$?
$$ \phi ( \phi(m)) = u \Big( v_1 \phi (m) v_1^* + v_2 \phi (m) v_2^* \Big) u^* $$