Given a language $L=\{E, =\}$ where $E$ is an equivalence relation.
Why does the statement $(\forall y)({\sim}Exy)$ have no quantifier-free formulation? Isn't: $$({\sim}Exa \mathbin\& {\sim}Exb \mathbin\& {\sim}Exc \ldots{\sim}Exn)$$ where $\{a, b, c,\ldots, n\}$ are all the objects in the universe, such a formulation?
According to my professor, the above statement has no quantifier-free formulation. Why isn't a simple conjunction sufficient? Because it could be infinite?
(I guess ignore that there is a $y$ such that $Exy$, namely $x$ itself?)
There are several potential issues here, and exactly which ones are relevant depends on the exact formalism you're using.
First, if your language is exactly $\{E,=\}$ then in the usual formulation of first-order logic, "$\sim Exa$" where $a$ is an object in your universe isn't actually a formula. (There are easy ways around this---for instance, adding a constant symbol for each object in your universe---so this may not be a serious obstacle in your set-up.)
Second, and this is the most important reason, talking about an equivalent formula often means the formula should be equivalent in all universes, not just in one. The formula you've written down is equivalent to $\forall y\ \mathop{\sim}Exy$ in that particular universe (assuming it's finite), but it won't work in any other universe.
Finally, as you noticed, if the universe is infinite, you'd need an infinite conjunction, which is no longer a first-order formula.