The Kanplan board is not available in the current server version, however, it does appear it will be available in the next version of JIRA which should be 7.4. This information is based on the Product Manager response to JSWSERVER-15236:
I'd like to inform you that we'll release Kanban Backlog functionality with the next JIRA Software version. We're now making some last polishing, heavily testing and opening an early access for our marketplace partners.
For those that know Kanban Backlog from JIRA Cloud you can expect a very similar set of functionalities with some improvements around specific use cases.
In order to deliver value early with the upcoming release we've decided to cut some functionalities from the original scope and what you might know from JIRA Cloud. This includes Epics and Versions filters panel on backlog that you know also from Scrum as well as new Release button.
Please see JSWSERVER-15236 for the full response from Jakub on April 6 2017. From the comments you can test the functionality in the 7.4 EAP release. If you plan on doing so please spin up a test server!
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