Why cosine is not an eigen signal?
That definition is rubbish.

An eigenfunction of a system is a function that passes that system, only scaled by a scalar value.

Now, suppose your system is a simple complex band pass filter, which has a pass band at $f_0$, but stop bands everywhere else, especially at $-f_0$.

Now your cosine $\cos(2\pi f_0 t)$ has spectral components at both $f_0$ and $-f_0$. The output of that filter contains only a multiple if the positive part. Hence, the output is not a scalar multiple of the input cosine. Done.