Вот решение, использующее CTE для установки ваших граничных условий.
-------------------------
-- TEST SCHEMA
create table #datatable (id varchar(6), emplrcd int, effdt date, [country visited] varchar (3))
insert into #datatable values
('112233',0,'10/9/2018 ','US '),
('112233',1,'10/10/2018','IND'),
('112233',1,'10/11/2018','BAN'),
('112233',1,'10/12/2018','PAK'),
('112233',0,'10/13/2018','US '),
('112233',2,'10/14/2018','IND'),
('112233',2,'10/16/2018','THA'),
('112233',2,'10/17/2018','SIN'),
('112233',0,'10/18/2018','US '),
('223344',0,'10/9/2018 ','US '),
('223344',1,'10/10/2018','IND'),
('223344',1,'10/11/2018','BAN'),
('223344',1,'10/12/2018','PAK'),
('223344',0,'10/13/2018','US '),
('223344',2,'10/14/2018','IND')
-------------------------
-- QUERY
; WITH boundryDate AS
(
SELECT
id
,min (effdt) mindate
, max(effdt) maxdate
FROM
#datatable
WHERE
[country visited] = 'US'
GROUP BY
id
)
SELECT
*
FROM
#datatable dt
JOIN boundryDate bd
on dt.id = bd.id
and dt.effdt between bd.mindate and bd.maxdate
WHERE
[country visited] <> 'US'