When does equality hold in this case? Give example of two vectors $x$ and $y$ such that $$||x+y||_2^2 = ||x||_2^2+||y||_2^2$$
and
$$<x,y>\neq0$$
I can't seem to find any two vectors $x$ and $y$ that satisfied both conditions at the same time.

If $x\perp y$, then $\|x+y\|^2=\|x\|^2+\|y\|^2$. 
E.g., for  $x,y \in \mathbb{R}^n$, 
$\|x+y\|^2=(x+y)^T(x+y)=x^Tx+2x^Ty+y^y=\|x\|^2+\|y\|^2+2*x^Ty$
if the dot product between the two is zero (i.e. $<x,y>=x^Ty=0$), then you have: $\|x+y\|^2=\|x\|^2+\|y\|^2$
