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I know that the eigenvalues for 4 point DFT matrix can be found from $F_4^4=I$. Is this also valid for 8, 16 and higher orders? For example with 8 points, will it be $F_8^8=I$ ? If not, how can I can compute them?

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The eigenvalues belong to the same set of quartic roots of unity verifying $\lambda^4=1$, whatever the order of the DFT.

For more details on their multiplicity, you can read: Eigenvectors and Functions of the Discrete Fourier Transform, 1982, Dickinson and Steiglitz (online).

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