Logged out users of a website hosted at WPEngine may receive cached responses to REST API calls via the GET method. API calls work correctly while logged in, but paths in the /wp-json/ directory are cached like much like the rest of the front-end.
How to prevent cached REST API responses
As of June 2017, the only way to disable this behavior is to open a ticket. WPEngine support technicians can create tickets based on live chat sessions.
Excluding a directory from caching happens at the install level, so a new request via ticket must be initiated after each new install is created.
UPDATE: WPEngine made this change to all 49 installs in my account in less than 24 hours.