Solution:

Such a number, which must consist entirely of $0$'s and $1$'s in base $3$, can never have more than one $1$. Indeed, if $n = 3^{a} + 3^{b} +$ higher powers where $b > a$, then $n^{2} = 3^{2a} + 2 \cdot 3^{a+b} +$ higher powers which will not be mild. On the other hand, if $n$ does just have one $1$ in base $3$, then clearly $n$ and $n^{2}$ are mild. So the values of $n \leq 1000$ that work are $3^{0}, 3^{1}, \ldots, 3^{6}$; there are $7$ of them.