Sumsets avoiding squarefree integers

We describe the structure of a set of integers $A$ of positive density $\delta$, such that $A+A$ contains no squarefree integer. It turns out that the behaviour changes abruptly at the values ${\delta }_0=1\mathrm{/}4-\frac{2}{{\pi }^2}=\mathrm{0.0473...}$ and ${\delta }_1=1\mathrm{/}18$.