Can the eigenvalue for a quantum mechanical operator be zero?
$S_z$ (spin along the z-axis) is a QM operator, and it frequently is zero - such as in closed-shell molecules.

Also note that the zero energy point of quantum mechanics can be defined in more than one way: some quantum chemistry programs use "all electrons and nuclei at infinite separation", whereas some solid state programs used "neutral atoms at infinite separation". Nevertheless, they evaluate the total energy.