There is a difference between the Traditional Square of Oppositions and the modern standard translation of so-called Categorical propositions.
According to Aristotle's Logic, the Universal Affirmative ($\text {Aab}$): "a belongs to all b" implies the corresponding Particular ($\text {Iab}$): "a belongs to some b".
With modern symbolization we have:
$\text {(Aab)} \ \ \forall x (b(x) \to a(x))$
and
$\text {(Iab)} \ \ \exists x (b(x) \land a(x))$.
The issue with so-called Existential Import is that $\text {Aab}$ implies $\text {Iab}$ under the condition that there are $\text A$'s.
The plausible explanation is that, according to Aristotle theory of science, terms are universals and they cannot be empty.
The origin of the Square is Aristotle's De Interpretatione (17b.17–26):
A and O are contradictories [they cannot both be true and they cannot both be false]; E and I are contradictories, and A and E are contraries [they cannot both be true but can both be false].
From this, the rest of the Square follows: I and O are subcontraries [they cannot both be false but can both be true], i.e. they cannot both be false, as well as subalternation [a proposition is a subaltern of another iff it must be true if its superaltern is true, and the superaltern must be false if the subaltern is false].
Subcontraries: suppose that I is false. Then its contradictory, E, is true. So E’s contrary, A, is false. An thus A’s contradictory, O, is true. The conclusion is that I and O cannot be both false.
Subalternation: suppose that A is true. Then its contrary E must be false. But then the E’s contradictory, I, must be true. Thus if the A form is true, so must be the corresponding I.
According to modern translation, $\text {Iab}$ is $\exists x (a(x) \land b(x))$ and $\text {Oab}$ is $\exists x (a(x) \land \lnot b(x))$.
If there are no $a$, both are false. Thus, according to modern point of view, two subcontraries can both be false.
