Probability

There are $36$ possible outcomes ($6$ for the first roll and $6$ for the second roll).

The only square numbers in the list are $1$ and $4$.

If we don't want either of these, there is $4$ sucsessful options for the first role and $4$ sucsessful options for the second roll. This means that there is $16$ succsessful outcomes, out of the $36$ total outcomes, meaning that the answer is $16\over 36$, which can be simplified to $\color{brown}\boxed{4\over9}$