The definition of Simple Harmonic Motion is :
simple harmonic motion is a type of periodic motion or oscillation motion where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement
From the definition of SHM we can state that, in order to be considered as SHM, a periodic motion only has to meet the following criteria that is:
(where $a$ is acceleration and $x$ is the displacement from equilibrium position) $$ a \propto -x$$ If we have a periodic motion where $$ x= \sin^2(\omega t)$$ $$\implies v=\frac{dx}{dt}=2\omega \sin(\omega t)\cos(\omega t)=\omega \sin(2\omega t)$$ $$\implies a=\frac{d^2x}{dt^2}=2\omega^2\cos(2\omega t)=2\omega^2(1-2\sin^2(\omega t))$$
Since, $x=\sin^2(\omega t)$, we get, $$a=2\omega^2(1-2x)$$ then, can we say that $$a \propto -x$$ and thus declare that it is a SHM?