Documentation for JIRA 5.2. Documentation for other versions of JIRA is available too. 
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The 'Assignee' field is required or automatically assigned by default when 'Allow unassigned issues' is turn OFF in Administration > General Configuration.
If it set as ON, user will be allow to choose the 'unassigned' option and this affect every project in JIRA. In certain use case, some project might need the Assignee field to be required and "unassign" is not an option.
In such case, there is not way to do it by default in JIRA, however this can be achieve by using a third party plugin call JIRA Suite Utilities , which has a set of conditions and validations to personalize workflow of JIRA.
To perform that: