Automating straightforward clinical tasks can reduce workload for healthcare professionals, increase accessibility for geographically-isolated patients, and alleviate some of the economic burdens associated with healthcare. A variety of preliminary screening procedures are potentially suitable for automation, and one such domain that has remained underexplored to date is that of structured clinical interviews. A task-specific dialogue agent is needed to automate the collection of conversational speech for further (either manual or automated) analysis, and to build such an agent, a dialogue manager must be trained to respond to patient utterances in a manner similar to a human interviewer. To facilitate the development of such an agent, we propose an annotation schema for assigning dialogue act labels to utterances in patient-interviewer conversations collected as part of a clinically-validated cognitive health screening task. We build a labeled corpus using the schema, and show that it is characterized by high inter-annotator agreement. We establish a benchmark dialogue act classification model for the corpus, thereby providing a proof of concept for the proposed annotation schema. The resulting dialogue act corpus is the first such corpus specifically designed to facilitate automated cognitive health screening, and lays the groundwork for future exploration in this area.