Isolate g(t) starting from $\int_{-\infty}^{+\infty}g^{2}(t)e^{-i2\pi f t}dt$
The integral is the Fourier transform of $g^2(t)$, so an inverse Fourier transform can recover $g^2(t)$. But without any further information you can't recover $g(t)$ from $g^2(t)$. So the problem is not the integral transform but simply the squaring operation.