Irreducibility of iterates of post-critically finite quadratic polynomials over $\mathbb {Q}$

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In this paper, we classify, up to three possible exceptions, all monic, post-critically finite quadratic polynomials $f(x)\in \mathbb {Z}[x]$ with an iterate reducible module every prime, but all of whose iterates are irreducible over $\mathbb {Q}$. In particular, we obtain infinitely many new examples of the phenomenon studied by Jones. While doing this, we also find, up to three possible exceptions, all integers $a$ such that all iterates of the quadratic polynomial ${(x+a)^2-a-1}$ are irreducible over $\mathbb {Q}$, which answers a question posed in by Ayad and McQuillan, except for three values of $a$. Finally, we make a conjecture that suggests a necessary and sufficient condition for the stability of any monic, post-critically finite quadratic polynomial over any field of characteristic $\neq 2$.