Is "Zero Shadow Day" necessary to perform Eratosthenes' experiment?

It's not necessary that there be zero shadow. The altitude $alpha$ of an object when it crosses the local meridian on the celestial sphere is $alpha = delta + 90^circ - lambda$, where $delta$ is the object's declination and $lambda$ is the latitude of the observation point. This means that if you carefully measure the altitude of the same object at meridian crossing from two separate locations, and find altitudes $alpha_1$ and $alpha_2$, then the difference between these altitudes is $$ alpha_2 - alpha_1 = delta + 90^circ - lambda_2 - (delta + 90^circ - lambda_1) = lambda_1 - lambda_2. $$ Thus, the difference in the measured altitudes is equal to the difference in latitude between the two locations. By measuring the distance between these locations on the Earth's surface, one can then infer the circumference of the Earth.

The condition that there be "zero shadow" simply means that one of $alpha_1$ or $alpha_2$ is equal to 90° (i.e, the sun is at the zenith.) Effectively, it allows you to know what one of the angles is without physically measuring the height of your gnomon and the length of its shadow. But so long as you're willing to measure the solar altitude in two different locations, it is not necessary that the sun be at the zenith.

Note that this method can be used to find the difference in latitude between any two points on the Earth, so long as the object's declination doesn't shift significantly between the meridian crossings at the two points. The reason it's helpful to have the observations at two points on the same longitude line is that it leads to a simple relation between the great-circle distance between the points, the difference in latitude, and the radius of the Earth. If the observation points aren't on the same line of longitude, then you need to know the difference in longitude as well to perform this experiment; and Eratosthenes didn't have Harrison's chronometer at hand.