Populating missing data based on previous month-end values

One approach to this problem is to do the following:

1. Emulate LEAD on SQL Server 2008. You can use APPLY or a suquery for this.
2. For rows without a next row, add one month to their account date.
3. Join to a dimension table that contains month end dates. This eliminates all rows that don't span at least a month and adds rows to fill in the gaps as necessary.

I modified your test data a little bit to make the results deterministic. Also added an index:

create table #histories
(
    username varchar(10),
    account varchar(10),
    assigned date  
);

insert into #histories 
values 
('PHIL','ACCOUNT1','2017-01-04'),
('PETER','ACCOUNT1','2017-01-15'),
('DAVE','ACCOUNT1','2017-03-04'),
('ANDY','ACCOUNT1','2017-05-06'),
('DAVE','ACCOUNT1','2017-05-07'),
('FRED','ACCOUNT1','2017-05-08'),
('JAMES','ACCOUNT1','2017-08-05'),
('DAVE','ACCOUNT2','2017-01-02'),
('PHIL','ACCOUNT2','2017-01-18'),
('JOSH','ACCOUNT2','2017-04-08'), -- changed this date to have deterministic results
('JAMES','ACCOUNT2','2017-04-09'),
('DAVE','ACCOUNT2','2017-05-06'),
('PHIL','ACCOUNT2','2017-05-07') ;

-- make life easy
create index gotta_go_fast ON #histories (account, assigned);

Here's the laziest date dimension table of all time:

create table #date_dim_months_only (
    month_date date,
    primary key (month_date)
);

-- put 2500 month ends into table
INSERT INTO #date_dim_months_only WITH (TABLOCK)
SELECT DATEADD(DAY, -1, DATEADD(MONTH, ROW_NUMBER() OVER (ORDER BY (SELECT NULL)), '20000101'))
FROM master..spt_values;

For step 1, there are plenty of ways to emulate LEAD. Here's one method:

SELECT 
  h1.username
, h1.account
, h1.assigned
, next_date.assigned
FROM #histories h1
OUTER APPLY (
    SELECT TOP 1 h2.assigned
    FROM #histories h2
    WHERE h1.account = h2.account
    AND h1.assigned < h2.assigned
    ORDER BY h2.assigned ASC
) next_date;

For step 2, we need to change the NULL values to something else. You want to include the final month for each account, so adding one month to the starting date suffices:

ISNULL(next_date.assigned, DATEADD(MONTH, 1, h1.assigned))

For step 3, we can join to the date dimension table. The column from the dimension table is exactly the column you need for the result set:

INNER JOIN #date_dim_months_only dd ON
    dd.month_date >= h1.assigned AND
    dd.month_date < ISNULL(next_date.assigned, DATEADD(MONTH, 1, h1.assigned))

I didn't like the query that I got when I put it all together. There can be issues with join order when combining OUTER APPLY and INNER JOIN. To get the join order I wanted I rewrote it with a subquery:

SELECT 
  hist.username
, hist.account
, dd.month_date 
FROM
(
    SELECT 
      h1.username
    , h1.account
    , h1.assigned
    , ISNULL(
        (SELECT TOP 1 h2.assigned
            FROM #histories h2
            WHERE h1.account = h2.account
            AND h1.assigned < h2.assigned
            ORDER BY h2.assigned ASC
        )
        , DATEADD(MONTH, 1, h1.assigned)
    ) next_assigned
    FROM #histories h1
) hist
INNER JOIN #date_dim_months_only dd ON
    dd.month_date >= hist.assigned AND
    dd.month_date < hist.next_assigned;

I don't know how much data you have so it might not matter for you. But the plan looks how I want it to:

The results match yours:

╔══════════╦══════════╦════════════╗
║ username ║ account  ║ month_date ║
╠══════════╬══════════╬════════════╣
║ PETER    ║ ACCOUNT1 ║ 2017-01-31 ║
║ PETER    ║ ACCOUNT1 ║ 2017-02-28 ║
║ DAVE     ║ ACCOUNT1 ║ 2017-03-31 ║
║ DAVE     ║ ACCOUNT1 ║ 2017-04-30 ║
║ FRED     ║ ACCOUNT1 ║ 2017-05-31 ║
║ FRED     ║ ACCOUNT1 ║ 2017-06-30 ║
║ FRED     ║ ACCOUNT1 ║ 2017-07-31 ║
║ JAMES    ║ ACCOUNT1 ║ 2017-08-31 ║
║ PHIL     ║ ACCOUNT2 ║ 2017-01-31 ║
║ PHIL     ║ ACCOUNT2 ║ 2017-02-28 ║
║ PHIL     ║ ACCOUNT2 ║ 2017-03-31 ║
║ JAMES    ║ ACCOUNT2 ║ 2017-04-30 ║
║ PHIL     ║ ACCOUNT2 ║ 2017-05-31 ║
╚══════════╩══════════╩════════════╝

Here I don't use calendar table but a natural numbers table nums.dbo.nums (I hope you've got it too, if not, it can be easily generated)

I have the answer slightly different from yours ('JOSH' <-> 'JAMES') because your data contains these 2 rows:

('JOSH','ACCOUNT2','2017-04-09'),
('JAMES','ACCOUNT2','2017-04-09'),

with the same account and assigned date and you did not precise which one should be taken is this situation.

declare @eom table(account varchar(10), dt date); 

with acc_mm AS
(
select account, min(assigned) as min_dt, max(assigned) as max_dt
from #histories
group by account
),

acc_mm1 AS
(
select account,
       dateadd(month, datediff(month, '19991231', min_dt), '19991231') as start_dt,
       dateadd(month, datediff(month, '19991231', max_dt), '19991231') as end_dt
from acc_mm
)

insert into @eom (account, dt) 
select account, dateadd(month, n - 1, start_dt)
from acc_mm1
      join nums.dbo.nums            
           on n - 1 <= datediff(month, start_dt, end_dt); 

select eom.dt, eom.account, a.username
from @eom eom 
     cross apply(select top 1 *
                 from #histories h 
                 where eom.account = h.account
                   and h.assigned <= eom.dt
                 order by h.assigned desc) a
order by eom.account, eom.dt;                          

Triangle JOIN for the win!

SELECT account,EndOfMonth,username
FROM (
    SELECT Ends.*, h.*
        ,ROW_NUMBER() OVER (PARTITION BY h.account,Ends.EndOfMonth ORDER BY h.assigned DESC) AS RowNumber
    FROM (
        SELECT [Year],[Month],MAX(DATE) AS EndOfMonth
        FROM #dim
        GROUP BY [Year],[Month]
        ) Ends
    CROSS JOIN (
        SELECT account, MAX(assigned) AS MaxAssigned
        FROM #histories
        GROUP BY account
        ) ac
    JOIN #histories h ON h.account = ac.account
        AND Year(h.assigned) = ends.[Year]
        AND Month(h.assigned) <= ends.[Month] --triangle join for the win!
        AND EndOfMonth < DATEADD(month, 1, Maxassigned)
    ) Results
WHERE RowNumber = 1
ORDER BY account,EndOfMonth;

Results are:

account     EndOfMonth  username

ACCOUNT1    2017-01-31  PETER
ACCOUNT1    2017-02-28  PETER
ACCOUNT1    2017-03-31  DAVE
ACCOUNT1    2017-04-30  DAVE
ACCOUNT1    2017-05-31  FRED
ACCOUNT1    2017-06-30  FRED
ACCOUNT1    2017-07-31  FRED
ACCOUNT1    2017-08-31  JAMES

ACCOUNT2    2017-01-31  PHIL
ACCOUNT2    2017-02-28  PHIL
ACCOUNT2    2017-03-31  PHIL
ACCOUNT2    2017-04-30  JAMES
ACCOUNT2    2017-05-31  PHIL

Interactive Execution Plan here.

I/O and TIME stats (truncated all the zero-values after logical reads):

(13 row(s) affected)

Table 'Worktable'.  Scan count 3, logical reads 35.
Table 'Workfile'.   Scan count 0, logical reads  0.
Table '#dim'.       Scan count 1, logical reads  4.
Table '#histories'. Scan count 1, logical reads  1.

SQL Server Execution Times:
    CPU time = 0 ms,  elapsed time = 3 ms.

Query to create required 'temp tables and test the T-SQL statement I'm suggesting:

IF OBJECT_ID('tempdb..#histories') IS NOT NULL
    DROP TABLE #histories

CREATE TABLE #histories (
    username VARCHAR(10)
    ,account VARCHAR(10)
    ,assigned DATE
    );

INSERT INTO #histories
VALUES
('PHIL','ACCOUNT1','2017-01-04'),
('PETER','ACCOUNT1','2017-01-15'),
('DAVE','ACCOUNT1','2017-03-04'),
('ANDY','ACCOUNT1','2017-05-06'),
('DAVE','ACCOUNT1','2017-05-07'),
('FRED','ACCOUNT1','2017-05-08'),
('JAMES','ACCOUNT1','2017-08-05'),
('DAVE','ACCOUNT2','2017-01-02'),
('PHIL','ACCOUNT2','2017-01-18'),
('JOSH','ACCOUNT2','2017-04-08'),
('JAMES','ACCOUNT2','2017-04-09'),
('DAVE','ACCOUNT2','2017-05-06'),
('PHIL','ACCOUNT2','2017-05-07');

DECLARE @StartDate DATE = '20170101'
    ,@NumberOfYears INT = 2;

-- prevent set or regional settings from interfering with 
-- interpretation of dates / literals
SET DATEFIRST 7;
SET DATEFORMAT mdy;
SET LANGUAGE US_ENGLISH;

DECLARE @CutoffDate DATE = DATEADD(YEAR, @NumberOfYears, @StartDate);

-- this is just a holding table for intermediate calculations:
IF OBJECT_ID('tempdb..#dim') IS NOT NULL
    DROP TABLE #dim

CREATE TABLE #dim (
    [date] DATE PRIMARY KEY
    ,[day] AS DATEPART(DAY, [date])
    ,[month] AS DATEPART(MONTH, [date])
    ,FirstOfMonth AS CONVERT(DATE, DATEADD(MONTH, DATEDIFF(MONTH, 0, [date]), 0))
    ,[MonthName] AS DATENAME(MONTH, [date])
    ,[week] AS DATEPART(WEEK, [date])
    ,[ISOweek] AS DATEPART(ISO_WEEK, [date])
    ,[DayOfWeek] AS DATEPART(WEEKDAY, [date])
    ,[quarter] AS DATEPART(QUARTER, [date])
    ,[year] AS DATEPART(YEAR, [date])
    ,FirstOfYear AS CONVERT(DATE, DATEADD(YEAR, DATEDIFF(YEAR, 0, [date]), 0))
    ,Style112 AS CONVERT(CHAR(8), [date], 112)
    ,Style101 AS CONVERT(CHAR(10), [date], 101)
    );

-- use the catalog views to generate as many rows as we need

INSERT #dim ([date])
SELECT d
FROM (
    SELECT d = DATEADD(DAY, rn - 1, @StartDate)
    FROM (
        SELECT TOP (DATEDIFF(DAY, @StartDate, @CutoffDate)) rn = ROW_NUMBER() OVER (
                ORDER BY s1.[object_id]
                )
        FROM sys.all_objects AS s1
        CROSS JOIN sys.all_objects AS s2
        -- on my system this would support > 5 million days
        ORDER BY s1.[object_id]
        ) AS x
    ) AS y;

/* The actual SELECT statement to get the results we want! */

SET STATISTICS IO, TIME ON;

SELECT account,EndOfMonth,username
FROM (
    SELECT Ends.*, h.*
        ,ROW_NUMBER() OVER (PARTITION BY h.account,Ends.EndOfMonth ORDER BY h.assigned DESC) AS RowNumber
    FROM (
        SELECT [Year],[Month],MAX(DATE) AS EndOfMonth
        FROM #dim
        GROUP BY [Year],[Month]
        ) Ends
    CROSS JOIN (
        SELECT account, MAX(assigned) AS MaxAssigned
        FROM #histories
        GROUP BY account
        ) ac
    JOIN #histories h ON h.account = ac.account
        AND Year(h.assigned) = ends.[Year]
        AND Month(h.assigned) <= ends.[Month] --triangle join for the win!
        AND EndOfMonth < DATEADD(month, 1, Maxassigned)
    ) Results
WHERE RowNumber = 1
ORDER BY account,EndOfMonth;

SET STATISTICS IO, TIME OFF;

--IF OBJECT_ID('tempdb..#histories') IS NOT NULL DROP TABLE #histories
--IF OBJECT_ID('tempdb..#dim') IS NOT NULL DROP TABLE #dim