LEFT JOIN alone will not cut it. You'll also have to move the WHERE condition to ON:

CREATE TABLE #StatusHistory (
 UniqueID varchar(30) ,
 Sequence int,
 BeforeStatus varchar(10),
 AfterStatus varchar(10),
 HistoryStatusDate date,
 PRIMARY KEY(UniqueID, Sequence)
 )

INSERT INTO #StatusHistory VALUES('08-40089445',1,'','BNT','20171017')
INSERT INTO #StatusHistory VALUES('08-40089445',2,'BNT','VFD','20180118')
INSERT INTO #StatusHistory VALUES('08-40089445',3,'VFD','NST','20180123')
INSERT INTO #StatusHistory VALUES('08-40089445',4,'NST','SLD','20180123')

SELECT * FRom
(SELECT s.*,h.HistoryStatusDate statusEnddate
FROM [#StatusHistory] S
LEFT JOIN [#StatusHistory] h
ON s.UniqueID=h.UniqueID
AND  s.Sequence=h.Sequence-1
) s
--Still trying to find the 7th record for this example
WHERE s.[UniqueID]='08-40089445'
ORDER BY s.UniqueID, s.Sequence
Result:
UniqueID                       Sequence    BeforeStatus AfterStatus HistoryStatusDate statusEnddate
------------------------------ ----------- ------------ ----------- ----------------- -------------
08-40089445                    1                        BNT         2017-10-17        2018-01-18
08-40089445                    2           BNT          VFD         2018-01-18        2018-01-23
08-40089445                    3           VFD          NST         2018-01-23        2018-01-23
08-40089445                    4           NST          SLD         2018-01-23        NULL

(4 rows affected)

However, I'd be reluctant to rely on the Sequence column being an unbroken sequence (the -1 condition). So I'd likely utilize an OUTER APPLY in stead of the LEFT JOIN:
SELECT * FRom
(SELECT s.*,h.HistoryStatusDate statusEnddate
FROM [#StatusHistory] S
OUTER APPLY (
 SELECT TOP(1) HistoryStatusDate
 FROM [#StatusHistory] h
 WHERE s.UniqueID=h.UniqueID
 AND h.Sequence > s.Sequence
 ORDER BY h.Sequence ASC
 ) h
) s
--Still trying to find the 7th record for this example
WHERE s.[UniqueID]='08-40089445'
ORDER BY s.UniqueID, s.Sequence
If we put in a 5 instead of 4 in the last row (thereby creating a gap in the Sequence column sequence), we can see the difference:
UniqueID                       Sequence    BeforeStatus AfterStatus HistoryStatusDate statusEnddate
------------------------------ ----------- ------------ ----------- ----------------- -------------
08-40089445                    1                        BNT         2017-10-17        2018-01-18
08-40089445                    2           BNT          VFD         2018-01-18        2018-01-23
08-40089445                    3           VFD          NST         2018-01-23        NULL
08-40089445                    5           NST          SLD         2018-01-23        NULL

(4 rows affected)

UniqueID                       Sequence    BeforeStatus AfterStatus HistoryStatusDate statusEnddate
------------------------------ ----------- ------------ ----------- ----------------- -------------
08-40089445                    1                        BNT         2017-10-17        2018-01-18
08-40089445                    2           BNT          VFD         2018-01-18        2018-01-23
08-40089445                    3           VFD          NST         2018-01-23        2018-01-23
08-40089445                    5           NST          SLD         2018-01-23        NULL

(4 rows affected)

Only the last one can cope.

Anyway, both methods will need an index on UniqueID, Sequence in order to perform adequately if more than a few rows are to be processed.