Additional values of Dedekind's $\eta$ function in radical form

Using this answer one can easily verify the value of $\eta(9i)$.

We have by definition $$\eta(9i)=e^{-3\pi/4}\prod_{n=1}^{\infty} (1-e^{-18n\pi})$$ And using the answer linked above we can see that $$\eta(9i)=\frac{\sqrt[3]{\sqrt[3]{18+6\sqrt{3}}-3}}{6}\cdot\frac{\Gamma (1/4)}{\pi^{3/4}}$$ One can use a little bit of algebra to verify that $$\sqrt{6}(2+\sqrt{3})^{1/6}=\sqrt[3]{18+6\sqrt{3}}$$ and get the value of $\eta(9i)$ mentioned in your question.

The modular equation given in the linked answer can be used to evaluate $\eta(27i)$ given the values of $\eta(3i),\eta(9i)$ and in general one can get the values of $\eta(3^ni)$ in similar fashion. Using $\eta(2i),\eta(6i)$ one can also verify the value of $\eta(18i)$. You should use the value of $\eta(7i)$ (given in linked question in your post) and $\eta(63i)$ of your post together with Ramanujan's modular equation to get the value of $\eta(21i)$ and add it to your table.

Remaining set of values of eta function in your question require more effort to verify.