Period of a continuous signal
Let $T_0$ be the period of the signal  $s(t) = x_1(t) + x_2(t)$.
$$s(t) = s(t+T_0) \tag{1}$$

Then the period for $s^2(t) = ( x_1 + x_2)^2$ will be either $T_0$ or $T_0/2$ :

First, it's obvious that $T_0$ is also a period for $s(t)^2$ as it satisfies $$s^2(t) = s^2(t+T_0) \tag{2}$$ 

However, $T_0 / 2$ may also be period for $s^2(t)$. Since we are looking for the smallest number to satisfy eq(2).

Hence do the following:


Find the period $T_0$ of $s(t) = x_1(t)+x_2(t)$.
Check if $T_0/2$ satisfies $s^2(t) = s^2(t+T_0)$.
If yes; then period of $s^2(t)$ is $T_0/2$; otherwise $T_0$.