In the paper by John Corcoran & Hassan Masoud (2014): Existential Import Today: New Metatheorems; Historical, Philosophical, and Pedagogical Misconceptions, History and Philosophy of Logic, already in the introduction it says, as self-evident, that
"The universalized conditional $∀x(x = 0 → x = (x + x))$ implies the corresponding existentialized conjunction $∃x(x = 0 \text { & } x = (x + x))$. And $∃x(x = 0)$ is tautological (in the broad sense, i.e. logically true)."
There are two assertions here, and I have difficulty with each one. In the first assertion I do not see why a model in which there is no $0$ (e.g., $\mathsf {ZFC^*}$ obtained by negating all the axioms of $\mathsf {ZFC}$), so that the conclusion of the implication is false and the premise would be (vacuously) true, would not be a counter-example. In the second assertion, I don't see why the statement is tautological (since it is not satisfied by all models, such as $\mathsf {ZFC^*}$).
As soon as you use a constant symbol, any interpretation will need to map that symbol to some object of its domain. So even if you say something like $\neg \exists x x=0$, you still need some object that $0$ denotes ... And since that object is of course identical to itself, this statement is a logical contradiction ... Meaning that $\exists x x=0$ is a tautology.
Something similar hold for the first one: whatever the $0$ denotes, it is of course identica to itself, and so by the universal it must then also be true that $0 = 0+0$, and hence the existential is true.