Assuming the law of excluded middle internally your formulation of Zorn's lemma is equivalent to the axiom of choice by the usual argument. Now there are Boolean Grothendieck topos in which the axiom of choice fails. Hence they don't satisfies Zorn lemma either.



I think the simplest example of this is to take $G$ to be a non-discrete but pro-discrete localic/topological group (typically $G = \mathbb{Z}_p$ the group of p-addic integer) and consider $\mathcal{T}$ the topos of sets endowed with a continuous (hence smooth) action of $G$ (that is the stabilizer of each point is an open subgroup)

Assuming LEM in the base, $\mathcal{T}$ is clearly boolean (sub-object are $G$-stable subsets and set theoretic complement preserve subobject).

But $\mathcal{T}$ doesn't satisfies either internal nor external choice. For exemple for $G = \mathbb{Z}_p$ the infinite product of all the $\mathbb{Z}/p^k\mathbb{Z}$ with their canonical action is the initial object. (more generally, the product of the $G/U$ for $U$ in a fundamental system of open subgroup is empty unless $G$ is discrete).


**Edit:** However it should be noted that an appropriate version of Zorn lemma is know to holds in every **Localic** Grothendieck topos (but the topos mentioned above is not localic of course). It says that every "inductie poset" has a maximal element but where "inductive" is defined as "every chain as a least upper bound".  It is proved in the paper by Bell linked in Peter Lumsdain comment, or it can be deduced from D4.5.14 in sketch of an Elepant (there are some technical points here, discussed in the comment below). It is unclear to me (and apparently to Bell when he wrote his paper) whether Zorn lemma where inductive is defined as "all chain have upper bounds" holds in localic topos.