1. An eigenvalue $\lambda$ of an operator $\hat{A}$ (with definite Grassmann-parity $|\hat{A}|$) is a complex [supernumber](http://planetmath.org/supernumber) of the same Grassmann-parity. See also [this](http://physics.stackexchange.com/q/40746/2451) Phys.SE post and links therein.

2. Note however that an annihilation operator $\hat{a}$ and a creation operator $\hat{a}^{\dagger}$ of definite Grassmann-parity $|\hat{a}|$ do not [supercommute](http://ncatlab.org/nlab/show/graded+commutator)
$$ [\hat{a}, \hat{a}^{\dagger}]~:=~\hat{a}\hat{a}^{\dagger}-(-1)^{\hat{a}}\hat{a}^{\dagger} \hat{a}~=~ \hbar~\hat{\bf 1}~\neq~0, $$
and are therefore not [supernormal](http://en.wikipedia.org/wiki/Normal_operator), and hence not diagonalizable, cf. e.g. [this](http://physics.stackexchange.com/q/82746/2451) Phys.SE post. OP's fermionic fields are field-theoretic versions of Grassmann-odd annihilation & creation operators.