From what I understand Euclid avoided infinity, and so I'm wondering how Euclid might have dealt with the concept of 0 in the Elements.
The arithmetic books of Elements are books VII - IX.
Euclid begins book VII with his definition of number:
A number is a multitude composed of units.
(This is taken from the Pythagorean notion of number.)
Since "zero" is not a multitude of units it does not satisfy Euclid's definition of number. Euclid did not consider "one" to be a number for the same reason. However, Euclid appears to have been somewhat ambivalent on this point and did treat the unit as a number on those occasions when it helped in stating a general proposition or proving them. For example, Euclid's famous proof (IX.20) that there are infinitely many prime numbers.