Signal squared in time domain
If somebody squares a real signal in the time domain, he probably deals with the signal's energy. Since this is a pointwise product, this results in a convolution in the Fourier domain: 

$$ \mathcal{F}(x\times x) = \mathcal{F}(x)\ast  \mathcal{F}(x) $$

In some cases, as stated by @MBaz, sometimes trigonometric identities can explain the frequency shift, as:


$\cos^2 x=\frac{1}{2}(1+\cos2x)$
$\sin^2 x=\frac{1}{2}(1-\cos2x)$


From the above formulae, one can see that:


they are positive, and the non-oscillatory term is $1/2 \cos(0x)$, explaining a peak at frequency zero,
the scalar before $x$ has doubled, explaining the peak moves.