We investigate the appearance of the square of a Hamilton cycle in the model of randomly perturbed graphs, which is, for a given $α\in (0,1)$, the union of any $n$-vertex graph with minimum degree $αn$ and the binomial random graph $G(n,p)$. This is known when $α> 1/2$, and we determine the exact perturbed threshold probability in all the remaining cases, i.e., for each $α\le 1/2$. We demonstrate that, as $α$ ranges over the interval $(0,1)$, the threshold performs a countably infinite number of `jumps'. Our result has implications on the perturbed threshold for $2$-universality, where we also fully address all open cases.