Thanks for all the input. Managed to get it to work.
So this time, it considers the year.
Run the code and you will see that 78576109 does not get counted ( He/she only has 2 months continuously )


--Select top 100 * FROM [EDW].[MEMBER].[MemberEligibilityByMonth] c where
--c.BeneficiaryID = '0068576102' 
If object_id('tempdb..#t') IS NOT NULL DROP TABLE #t;
CREATE TABLE #t( BeneficiaryID VARCHAR(10), EligYear INT, EligMonth INT );

INSERT INTO #t(BeneficiaryID, EligYear, EligMonth )
Select '0068576102',2016,3 UNION
Select '0068576102',2016,6 UNION
Select '0068576102',2016,7 UNION
Select '0068576102',2016,8 UNION
Select '0068576102',2016,9 UNION
Select '0068576102',2016,10 UNION
Select '0068576102',2016,11 UNION
Select '0068576102',2016,12 UNION
Select '0068576102',2017,10 UNION
Select '0068576102',2017,11 UNION
Select '0068576102',2017,12 UNION
Select '0078576103',2016,8 UNION
Select '0078576103',2016,9 UNION
Select '78576103',2016,8 UNION
Select '78576103',2016,9 UNION
Select '78576103',2016,10 UNION
Select '78576109',2016,8 UNION
Select '78576109',2016,9 UNION
Select '78576109',2016,11 UNION
Select '78576109',2016,12

;
WITH T AS
(
SELECT *
     ,DENSE_RANK() OVER (PARTITION BY BeneficiaryID, EligYear ORDER BY EligMonth) - EligMonth AS Grp2
   --,DENSE_RANK() OVER (ORDER BY EligMonth) - EligMonth AS Grp
FROM #t
)
--Select * FROM T ORDER BY 1
,
CONSECUTIVE_MONTHS_TOGETHER as
(
SELECT
     BeneficiaryID,
     EligYear,
     MIN(EligMonth) AS RangeStart,
   MAX(EligMonth) AS RangeEnd,
     MAX(EligMonth) - MIN(EligMonth) as DIFF
FROM T
GROUP BY BeneficiaryID,EligYear, Grp2
)
Select * FROM CONSECUTIVE_MONTHS_TOGETHER WHERE DIFF >= 2 --( See Explanation )

/*
This solution works when the numbers are in a sequence ( that is the case with my example )
So if you had any 3 months of consecutive coverage the difference between the largest and smallest must be > 1 -- OR ( >= (2) )
*/