SELECT MemberID, MemberName, ReportingQuarter, MIN(FinalExpectedDay) AS EarliestCompletionDay
FROM (
    SELECT MemberID, MemberName, ReportingQuarter, FinalExpectedDay, 
        ROW_NUMBER() OVER (PARTITION BY MemberID, ReportingQuarter ORDER BY FinalExpectedDay) AS RowNum,
        COUNT(*) OVER (PARTITION BY MemberID, ReportingQuarter) AS TotalRows
    FROM YourTable
) t
WHERE RowNum >= CEILING(TotalRows * 0.6)
GROUP BY MemberID, MemberName, ReportingQuarter

Should work for 60%

You can do it by divide the row number by count to get the Percentile :

select *, CAST(rn as float)/CAST(count as float)*100 as Percentile
from (
  select MemberID, MemberName, ReportingQuarter, FinalExpectedDay,
         ROW_NUMBER() OVER (PARTITION BY MemberID ORDER BY FinalExpectedDay) as rn,
         count(*) over (partition by MemberID) as count
  from mytable
) as s
where CAST(rn as float)/CAST(count as float)*100 = 60

Demo here

I converted your example data into easily reproducible DDL/DML. This is really helpful when asking questions like this:

DECLARE @table TABLE (MemberID INT, MemberName NVARCHAR(20), ReportingQuarter DATE, FinalExpectedDay DATE);
INSERT INTO @table (MemberID, MemberName, ReportingQuarter, FinalExpectedDay) VALUES
(1, 'A', '12/31/2022', '3/21/2023'), (1, 'A', '12/31/2022', '3/22/2023'), (1, 'A', '12/31/2022', '3/23/2023'),
(1, 'A', '12/31/2022', '3/24/2023'), (1, 'A', '12/31/2022', '3/25/2023'), (2, 'B', '12/31/2022', '3/21/2023'),
(2, 'B', '12/31/2022', '3/22/2023'), (2, 'B', '12/31/2022', '3/23/2023'), (2, 'B', '12/31/2022', '3/24/2023'),
(2, 'B', '12/31/2022', '3/25/2023');

;WITH allData AS  (
        SELECT MemberID, MemberName, ReportingQuarter, FinalExpectedDay, 
               COUNT(*) OVER (PARTITION BY MemberID ORDER BY (SELECT 1)) AS Total, COUNT(*) OVER (PARTITION BY MemberID ORDER BY FinalExpectedDay) AS RunningTotal,
               (COUNT(*) OVER (PARTITION BY MemberID ORDER BY FinalExpectedDay)+.0) / (COUNT(*) OVER (PARTITION BY MemberID ORDER BY (SELECT 1))+.0) AS PctRunningTotal
          FROM @table
)

SELECT sixty.MemberID, sixty.MemberName, sixty.FinalExpectedDay AS SixtyExpectedDate, eighty.FinalExpectedDay AS EightyExpectedDate, ninty.FinalExpectedDay AS NintyExpectedDate
  FROM allData sixty
    INNER JOIN allData eighty
      ON sixty.MemberID = eighty.MemberID
      AND eighty.PctRunningTotal BETWEEN .8 AND .8999
    INNER JOIN allData ninty
      ON sixty.MemberID = ninty.MemberID
      AND ninty.PctRunningTotal >= .9
 WHERE sixty.PctRunningTotal BETWEEN .6 AND .7999

Here we're using a CTE to do the figuring out for us. We determine the total count per member, and the percent completed for each date.

Then we select from it, first for the sixty percent value, then join back to it for the eighty percent value and finally again for the ninty.
| MemberID | MemberName | SixtyExpectedDate | EightyExpectedDate | NintyExpectedDate |
| ------------ | ------------ | ------------ | ------------ | ------------ |
| 1 | A | 2023-03-23 | 2023-03-24 | 2023-03-25 |
| 2 | B | 2023-03-23 | 2023-03-24 | 2023-03-25 |