Spectral effects of multiplying a signal by itself?
Is squaring a signal different than multiplying it by itself 


No, it is not different.  Squaring the signal is multiplying it by itself, sample by sample.

In the frequency domain this is equivalent to convolving the original frequency spectrum with itself.  The convolution "spreads" the signal out.  If the signal is at $\pm300$ Hz around a carrier frequency $f_c$, then you should see energy at $\pm(2f_c + 600)$ Hz, $\pm2f_c$ Hz, $\pm(2f_c - 600)$ Hz, $0$ Hz, and $\pm600$ Hz.