I am interested in the classifying space $BG$ of a finite group $G$.

A real representation $V$ of $G$ of dimension $r$ defines a real vector bundle over $BG$ of rank $r$. If the determinant of this representation is trivial, then this bundle is orientable and a choice of orientation determines an Euler class $e(V) \in H^r(BG,\mathbb{Z})$.

Do these classes generate $H^*(BG,\mathbb{Z})$ as a ring?

This is easy to show when $G$ is abelian. In that case we need only consider $G = \mathbb{Z}_n$ and the cohomology ring is generated in degree 2 by the Euler class of the $2\pi/n$-rotation representation.

More generally, for any $G$, the map $H^1(BG,SO(2)) \to H^2(BG,\mathbb{Z})$ is an isomorphism since the latter is torsion and this is a Bockstein-type operation. We can consider the element of $H^1(BG,SO(2))$ as a homomorphism $G \to SO(2)$ giving us a 2d representation and it is easy to see from the definition of the Bockstein that the above map is the Euler class of this representation.