In ZFC, a class is given by a class formula $\phi$, that is, a formula expressed in the language of ZFC. For example, the notion of group can be formalized by such a formula $\phi$. The notion of group homomorphism can also be so formalized, say by $\psi$. So in principle there is no problem in being able to write down a formula in ZFC whose abbreviation is 

$$\exists 1 \ \forall G \ \phi(G) \Rightarrow \exists ! f \ \psi(f) \wedge dom(f) = G \wedge cod(f) = 1$$ 

and this is what you're after. 

"Unbounded" quantification takes place all the time in ZFC. For example, the pairing axiom involves the formation of a set $\{x, y\}$ for any two sets $x, y$, and this "any" involves quantification over the class of sets. You do of course have to be careful around unbounded quantification, as we know from the classical set-theoretic paradoxes, but this is well worked out.