Can a system have negative entropy?

The entropy $S$ of a system is related to the number of possible microstates $Omega$ that a system can adopt in the following way:

$$S = k_B log Omega$$

Note that $Omega$ must always be an integer, and it must always be at least 1; hence, $S$ is always greater than or equal to zero.

In the zero-entropy case, the object is a perfect crystal at zero temperature, which has only one possible microstate. (Thus, the above definition is made possible by the Third Law of Thermodynamics.) Any other situation has more than one possible microstate, so the entropy must be greater than zero.

I think what you mean is that the entropy doesn't change for reversible processes, but increases for irreversible processes. In this sense your question would be if the entropy of a system can decrease. Yes, absolutely! The entropy can decrease for a system that is not closed. For example, Earth receives the solar energy prom the Sun and dissipates it into space as heat. The entropy of the whole (closed) system (Sun, Earth, and space) always increases. However, the entropy on Earth alone can indeed decrease. Entropy is often referred to as a measure of chaos, so order would be the opposite of entropy. In this sense biological life and evolution represent a highly organized matter and therefore a low entropy. Such a reduction of the entropy as the emergence of life and its evolution on Earth was possible exactly because Earth alone is not a closed system, but a conduit of a tremendous entropy increase of the solar energy dissipating as heat. Without this constant entropy increase, life on Earth would be impossible. It is exactly the entropy increase in the entire system that allowed the entropy in the part of the system to decrease thus producing life, evolution, and ultimately intelligence.