Are vectors in the null space of a matrix considered eigenvectors? From what I've learned about the definition of an eigenvector, it seems like a vector that gets mapped to zero should just be considered an eigenvector where $\lambda = 0$. Is that true, or are those considered a special case?

Yes it is correct by the definition, for $\vec x\neq 0$
$$A\vec x=0\vec x$$
then $\vec x$ is an eigenvector with eigenvalue $\lambda=0$.
