Nyquist Rate (Sampling Frequency) for $ {f}^{2} \left( x, y \right) $
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.