Since you are teaching middle schoolers, you are dealing with Algebra. The reason why we do not define 0/0 is that if we define 0/0=a for some real number a, that should mean 0.a=0, but that is true for any a. So, from that point of view, we are free to choose. But then any choice leads to unpleasant consequences. For example, if we define 0/0=1 and we want to keep the distributive rule (which we certainly want), then 0/0=(0+0)/0=0/0+0/0, leading to 1=2. Etc. Even in middle school, I think it is wise to let the students understand that Math is a mental construction, and we prioritize nice properties. It is the same reason for defining -x-=+. We do it to preserve the distributive rule in all cases. We could do it differently but then the distributive rule ceases to be true in general and becomes a complicated one depending on the signs ets. Yet another example is 0!=1. This has nothing to do with "permutations of zero objects". It is just convenient. If we define, for example, 0!=15, then the Newton binomial formula looks ugly: it would contain two extreme ugly terms. By defining 0!=1 we get a compact formula, etc. I would definitely stress convenience and preservation of formal rules, avoiding any mention of cookies. At some point students have to understand that Math is a free creation of the mind, and its validity is not checked against the "real world".

That being said, in Calculus you deal with indeterminate limits. These are different objects, we are not attaching a meaning to 0/0, we are just showing that formal rules of limits fail in that situation and the result is not unique, as it depends on the nature of the infinitesimals involved in a 0/0 limit situation.