The group $\langle x,y\mid x^2,y^2\rangle$ is the infinite dihedral group $D_\infty$.
In this group, $x$ and $y$ are distinct elements. One way to see this is to notice that there is a group homomorphism $\phi:D_\infty\to E$ where $E$ is the group of euclidean motions of the line, such that $$\phi(x):t\in\mathbb R\mapsto -t\in\mathbb R$$ and $$\phi(y):t\in\mathbb R\mapsto 1-t\in\mathbb R.$$ Indeed, once you show that $\phi$ exists, then obviously $\phi(x)\neq\phi(y)$, so $x\neq y$.
If $G$ is a group generated by two elements of order two, then there exists a surjection $D_\infty\to G$, so what you want to do is to classify all normal subgroups of $D_\infty$. To do this, it is useful to observe that the element $z=xy$ has infinite order in $D_\infty$ (this is obvious if you see what $\phi(z)$ is) and then show that a non-trivial normal subgroup must contain a power of $z$.