>>Can the eigenvalue of this operater tend to negative infinity?

Yes -- actually this question is closely tied to the physical origins of the Dirac operator.  Very roughly speaking negative eigenvalues correspond to negative-energy eigenstates; the Dirac equation predicts that particles with positive energy (e.g. electrons, which can have arbitrarily large positive energy) there is an antiparticle "twin" with negative energy (positrons in this case).

For instance, if $M$ is the unit sphere in $\mathbb{R}^3$ then Dirac eigenfunctions give you the angular part of the electron wave function; the corresponding eigenvalues are simply all the integers $n \in \mathbb{Z}$ which show up with multiplicity $n+1$.