I believe you can achieve this by setting the permissions from all projects in the board's filter. Basically, you need to set the "Browse Projects" permission to "Anyone", so no one will need to log into your system to see the project's information (and the board related to it).
Hope it helps!
Coincidentally, I just created a knowledge document explaining how to make an Agile board visible to anyone that have access to your JIRA instance (without the need to login and, therefore, use a license).
However, without public access, there's a need to use a license even if the user only has read-only permissions. An option would be to use HTTP authentication ('htaccess', for example) methods to block access in a page-level manner.
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