Let $x[n]$ be aliased sampling of real-valued $x(t)$ over $t_0 \leq t \leq t_1$. Can $|x(t)|^2$ be recovered more accurately than $x(t)$, over $t_0 \leq t \leq t_1$? If so, how?
For $|x[n]|^2$, superficially it's "yes", as aliasing can circularly shift certain $X[k]$, which $|x[n]|$ is invariant to. For $|x(t)|^2$, I see potential as it's lower in information.
Feel free to assume the amount of aliasing to be "limited", by whatever amount necessary (but do specify it).
"Standard theory", meaning sinc interpolation, says "no", and that much is clear. All modulus is aliased, since it's many-to-one. Yet this same flaw is a strength in the general framework, as $|x|$ is lower-information, hence inherently takes less to recover. For example, synchrosqueezed STFT can beat sinc interpolation for non-uniform sampling.
For complex $x$ that's a bandpass signal, $|x|$ can often be recovered more accurately, as modulus heavily shifts energy toward lower frequencies, even if new higher ones are introduced. Different with real $x$, but case in point, I'd not conclude it can't be done just because sinc can't do it.