Nyquist Rate (Sampling Frequency) for $ {f}^{2} \left( x, y \right) $ We are given that $f(x,y)$ is highest frequency is $\omega$ what will be the frequency sample rate if we want to restore the function of the form $g(x,y)=f^2(x,y)$ 

Would it be correct to say that because $\sin^2x=1-\cos 2x$, we will have to sample it at $4\pi$ ?
By the Convolution Theorem multiplication in Time / Spatial domain is equivalent of Convolution in the Frequency Domain.

The sampling rate (In its classic interpretation) is proportional to the support of a function in the frequency domain.

So if a function has a certain support in frequency, what would be its support after convolution with itself? Indeed it will be doubled in each dimension.