What is a combinatorial proof for the Fibonacci identity $F^2_{n+1} - F^2_{n-1} = F_{2n}$?

Question

I've been working on this problem for a while, trying to break it down in terms of the number of ways you can fill a board with squares and dominoes.

I know that for the similar identity $F^2_{n} + F^2_{n-1} = F_{2n-1}$, you can construct a $1 \times2n$ board where you can count it one way as just $F_{2n}$, and the other way you can count it by splitting the board into 2 $1 \times n$ boards, where you can add together all the possible ways you can fill each half assuming a square in the center ($F^2_{n}$), or assuming a domino in the center ($F^2_{n-1}$).

But for this identity, I don't know how to intutively explain taking away the options of a $F^2_{n-1}$ from the options of a $F^2_{n+1}$ board to get the options for a $F_{2n}$ board.

Answer 1

This is Identity 14 in Proofs that Really Count: The Art of Combinatorial Proof. The hint is:

Count the number of pairs of $n$-tilings where at least one ends in a square. Condition on whether the first tiling ends in a square or not.