Proceed with the following steps to disable the unassigned option:

1. Log into Jira with a user that has a Jira administrator's permission.
2. Navigate to Administration > System.
3. Click on the Edit Settings button.
4. Scroll down and select the OFF option for Allow unassigned issues.
5. Click on Update to apply the change.

Have a look at the following screenshot:

As you can see, once the option is disabled, issues can no longer be unassigned.