I have seen the proof of $\sum_{k=1}^{n} {{k} {n \choose k}^2 ={ n} {{2n-1} \choose {n-1}}}$ done with boys and girls and I somehow understand it, but $$\sum_{k=0}^{n}k^2 {n \choose k}^2 = n^2 {2n - 2 \choose n- 1}$$
is a little different.
I tried it like this:
What is the number of ways a group of $n$ boys and $n$ girls can be divided into a team of $n$ people with a boy leader and a girl supervisor who isn't a part of the team?
On the RHS:
There are two groups of n boys and n girls and we choose the leader from the $n$ boys and then we choose the supervisor from the $n$ girls and then we choose a team of $n - 1$ people for the boy from $2n - 2$ children.
On the LHS:
We can rewrite it into
$$\sum\limits_{k=0}^nk{n \choose k}k{n \choose n - k}$$
We first choose $k$ boys for a team and then a leader from them and then we choose girls who aren't part of the team. And I don't know what's next.