Robert Gerbicz wrote: Let $ n>2$ be a positive integer, then from the definition: \[ x_{n}=1+{x_{1}}^{2}+\cdots+{x_{n-2}}^{2}+{x_{n-1}}^{2}=x_{n-1}+{x_{n-1}}^{2}\] This is also true for $ n=2$. So $ x_{n}$ is between two square numbers: \[ {x_{n-1}}^{2}<x_{n}={x_{n-1}}^{2}+x_{n-1}<{x_{n-1}}^{2}+2*x_{n-1}+1=(x_{n-1}+1)^{2}\] proving the problem.

Notice that for all $n>1$, $x_n$ is even. Thus, $x_n\equiv 1+1+0+0+\cdots\equiv 2 \pmod 4$. Thus, $x_n$ can never be a perfect square.