How to do a FULL OUTER JOIN in SQLite?

Yes. See the Join (SQL) > Outer join > Full outer join article on Wikipedia.

SELECT employee.*, department.*
FROM   employee 
       LEFT JOIN department 
          ON employee.DepartmentID = department.DepartmentID
UNION ALL
SELECT employee.*, department.*
FROM   department
       LEFT JOIN employee
          ON employee.DepartmentID = department.DepartmentID
WHERE  employee.DepartmentID IS NULL