Você pode usar correlacionado subconsultas:
SELECT t1.case_id, t1.assigned_date,
(SELECT t2.assigned_date
FROM mytable AS t2
WHERE t2.case_id = t1.case_id AND
t2.assigned_date > t1.assigned_date
ORDER BY t2.assigned_date LIMIT 1) AS next_date,
(SELECT t2.assigned_date
FROM mytable AS t2
WHERE t2.case_id = t1.case_id AND
t2.assigned_date < t1.assigned_date
ORDER BY t2.assigned_date DESC LIMIT 1) AS prev_date
FROM mytable AS t1