Square Fibonacci numbers

Question

Are there Fibonacci numbers other than $F_0 = 0 = 0^2, F_1 = F_2 = 1 = 1^2,$ and $F_{12} = 144 = 12^2$ which are square numbers? If not, what is the proof?

Answer 1

Here is a paper of Bugeaud, Mignotte, and Siksek proving that

the only perfect powers in the Fibonacci sequence are 0, 1, 8, 144

Therefore the only squares are 0, 1, and 144.

Answer 2

0, 1 and 144 are the only perfect squares. See the proof here.