In quantum field theory, the deviations of the field operators from the vacuum expectation value must be linear combinations of creation and annihilation operators satisfying, like the fields themselves, canonical commutation or anticommutation relations. The fields therefore have the form VEV + such a linear combination.
Working out commutation relations gives no restriction on the VEV, but working out anticommutation relations leads to a contradicition unlesc the VEV vanishes. Therefore bosonic fields may have nonvanishing VEV, but for fermionic fields the VEV must vanish.
The same holds in any other state for the expectation values; only the creation and annihilation operators change, being related by a Bogoliubov transformation to those of the vacuum.