$L=\{a^ib^i|i\geq0\}$, cfg for $L^2$
The square language $L^2$ is defined as follows:
$$
L^2 = LL = \{ xy : x,y \in L \}.
$$
There is no requirement that $x = y$. Indeed, in your case $L^2 \supsetneq \{ w^2 : w \in L \}$.