First you need to ask yourself what is cardinality? It's a mathematical notion which is designed to give us a way to measure the size of a set.
When you want to extend cardinals to negative integers you need to ask yourself, what is the purpose of this extension? Can we measure "more sets" now? In the context of $\sf ZFC$ or similar theories the answer is no. You don't have any sets which can be measured by negative cardinals because you wish to extend the current system, where every set is already assigned a positive (or infinite) cardinal.
What about other set theories? Well, maybe there you could extend this context. But then you have a question of usability. Cardinality comes with a wonderful set of arithmetical operations. We know that $|A\times B|=|A|\cdot|B|$ and $|A+ B|=|A|+|B|$ (where $A+B$ is the disjoint union of the sets).
Can we extend these notions of arithmetics as well? If $|A|=-1$ then $|A\times A|=1$, and therefore $|A+A\times A|=0$. So we have a disjoint union of two non-empty sets is empty, or that cardinality zero is no longer the unique property of the empty set. Either case is troubling.
This means that cardinality, when extended to negative integers, cannot extend the usual arithmetics as well.
The last resort, if so, would be to measure a whole other object. Polytopes, for example. But this sort of misses the point that cardinality is the size of sets. So you can talk about the concept of "generalized cardinality" but that won't be a measure of size of sets anymore. What would characterize it exactly? I don't know. I just feel that the term "cardinality" wouldn't be very fitting here.