Is there an Elementary Theory of the Category of Groups?

Yes : see this paper (in French) by Pierre Leroux.

I'm not very familiar with the paper, but the main result is that a category $\mathcal{C}$ is equivalent to the category of groups if and only if :

1. It is complete, cocomplete and pointed.
2. There exists an object $G$ that is a regular projective generator.
3. This object is also a co-group, i.e. a group object in $\mathcal{C}^{op}$.
4. The full subcategory of $\mathcal{C}$ whose objects are the sums of isomorphic copies of $G$ is generated as a category with coproducts by the comultiplication $G\to G+G$, the inverse $G\to G$ and the zero map $G\to 0$.
5. Every prenormal subobject (i.e. an object that is the equivalence class of $0$ for an internal equivalence relation) is normal (i.e. a kernel).
6. Every subobject of a free object is free (where a free object is one that is isomorphic to $E\cdot G$ for some set $E$).
7. Every object is a subobject of a simple object (where a simple object is an object without non-trivial normal subobjects).

The first 4 conditions are condensed in the paper by saying that $(\mathcal{C},G)$ is a "projective category of groups" ("catégorie projective de groupes").

To summarize the proof : the first 4 conditions are equivalent to the fact that $\mathcal{C}$ is equivalent to a subcategory $\mathcal{G}$ of $\mathbf{Grp}$ closed under subobject and products. The fifth one implies $\mathcal{G}$ is also closed under quotients, and is thus a subvariety of the variety of groups. The only subvarieties where every subobject of a free object is free (these are called Schreier varieties) are the variety of abelian groups and the varieties of abelian groups of exponent $p$ for $p$ prime, and these do not have the seventh property because they have only $0$ or $1$ simple object (depending on the case).