Set $a = x_1^2 + y_1^2$. If $a$ is odd, the numbers $x_i$ and $y_i$ do not have the same parity, so $x_i + y_i$ is odd. Since $\sum (x_i + y_i) = 0$, it follows that $n$ is even.

Suppose $a = 4k+2$. Then $x_i$ and $y_i$ are both odd. The equality $x_1+x_2+\dots+x_n = 0$ implies $n$ is even.

Finally, if $a = 4k$, then $x_i$ and $y_i$ are even. The numbers $a_i = \frac{x_i}{2}$ and $b_i = \frac{y_i}{2}$ satisfy the initial conditions. Furthermore, $a_1^2 + b_1^2 = \frac{a}{4}$. Repeating the argument for $\frac{a}{4}$ instead of $a$, after a finite number of steps we end up in a previous case.