The miscommunication between the OP and her interlocutor is an example of what sometimes happens in interaction between people whose ways of speaking are shaped by an education or professional experience that revolves around analysing phenomena in quantitative terms, and those with other kinds of backgrounds. The differences that the latter group characterises as qualitative, may be quite spontaneously characterised as quantitative by the former. (So, no, the OP is far from being the only person to have had the experience of such miscommunication.)
Consider, for example, the difference between moving and standing still. To many people that is probably a very clear, definite, qualitative difference. Such people may speak of moving things as having this or that speed, but would never speak of the speed of an object that is standing still. A scientifically trained person, on the other hand, finds it quite natural to say that such an object has the speed that equals zero. In such a person’s conceptual framework, the difference between moving and standing still is merely quantitative; it’s the difference between having the speed of zero and having some other speed. A person who is accustomed to this way of thinking may feel compelled, when speaking of things that are in fact moving, to say that they have some non-zero speed; to people on the other side, ‘non-zero’ in such a context seems redundant, as they would never apply the concept of speed to motionless things.
Now, the same division can be seen when people speak of probabilities. The everyday framework for conceptualising them has the concepts such as impossible, possible (but improbable), probable (likely), certain. The differences among these at first appear to be qualitative, and are spoken of as such. People who are trained to analyse probabilities in quantitative terms, however, think of them as a continuum between zero and one. In that framework, something that is impossible has the probability of zero, something that is certain has the probability of one, and everything that is possible but not certain has some probability that is between these extremes. A person who is accustomed to that framework may feel the need to use the phrase ‘non-zero probability’ or ‘non-zero chance’ to make it clear that whatever is talked about is not impossible. To a person who is not accustomed to it, such a phrase seems strange, just like the non-zero speed in the above example. (Incidentally, to answer directly the question posed in the title, yes, non-zero in this context means more than zero as the scale of probabilities does not go below zero.)
So, saying that something has a non-zero chance is just a way of saying that it is possible, that comes naturally to people who have a certain educational or professional background, but may be confusing to those who don’t. Contrary to what the OP suspected, this way of speaking is not peculiar to Americans or to speakers of any other regional variation of the language. In fact, the whole matter is not specific to English, as analogous differences between people of different educational/professional backgrounds can probably be found among speakers of any language.
Now, the miscommunication between the OP and her interlocutor had another layer that was superimposed on this. The term non-zero chance, just like the more mundane term possible, by its meaning covers a wide range, from the probabilities that are just a sliver above zero, all the way to one. However, it would be strange and misleading (but not false) to use either of these terms if one knew that the probability is very high. Although these terms do not logically entail that the probability is low, they do imply it (in the loose, everyday sense of imply), or implicate it, or suggest it. In other words they convey the idea that the probability is low, as a matter of pragmatics, but not as a matter of semantics.