Why this system is linear?
That's not an integral of variable $x$. The notation $x(a+T/2)$ stands for a function $x(\cdot)$ of variable $a$.
So applying the fundamental theorem of calculus, and assuming there exists a function $G(a)$ such that $G'(a) = x(a)$, then you will have :
$$ \int x(a+T/2) da = \int G'(a+T/2)da = G(a+T/2) + C $$
where the constant of integration, $C$, will be omitted in the definite integral :
$$ \int_{t+T}^{t-T/2} x(a+T/2) da = \int_{t+T}^{t-T/2} G'(a+T/2)da = G(a+T/2)|_{t+T}^{t-T/2}  $$
So the system has nothing with a square function.
Coming to its linearity, you can show this in line with the linearity of the integral operator...