Intent classifiers must be able to distinguish when a user's utterance does not belong to any supported intent to avoid producing incorrect and unrelated system responses. Although out-of-scope (OOS) detection for intent classifiers has been studied, previous work has not yet studied changes in classifier performance against hard-negative out-of-scope utterances (i.e., inputs that share common features with in-scope data, but are actually out-of-scope). We present an automated technique to generate hard-negative OOS data using ChatGPT. We use our technique to build five new hard-negative OOS datasets, and evaluate each against three benchmark intent classifiers. We show that classifiers struggle to correctly identify hard-negative OOS utterances more than general OOS utterances. Finally, we show that incorporating hard-negative OOS data for training improves model robustness when detecting hard-negative OOS data and general OOS data. Our technique, datasets, and evaluation address an important void in the field, offering a straightforward and inexpensive way to collect hard-negative OOS data and improve intent classifiers' robustness.