cómo calcular el número total de sábado y domingo entre dos fechas en mysql

Si no tiene una tabla que tenga todas las fechas entre la fecha de inicio y la fecha de finalización (incluidas), entonces debe adoptar una consulta que cree primero todas las fechas entre el intervalo de fechas dado (incluidas). Luego usa WEEKDAY función de MySQL para comprobar si el día es Saturday o Sunday

SELECT 
dateTable.Day,
DAYNAME(dateTable.Day) AS dayName
FROM 
(   SELECT ADDDATE('2016-07-01', INTERVAL @i:[email protected]+1 DAY) AS DAY
    FROM (
        SELECT a.a
        FROM (SELECT 0 AS a UNION ALL SELECT 1 UNION ALL SELECT 2 UNION ALL SELECT 3 UNION ALL SELECT 4 UNION ALL SELECT 5 UNION ALL SELECT 6 UNION ALL SELECT 7 UNION ALL SELECT 8 UNION ALL SELECT 9) AS a
        CROSS JOIN (SELECT 0 AS a UNION ALL SELECT 1 UNION ALL SELECT 2 UNION ALL SELECT 3 UNION ALL SELECT 4 UNION ALL SELECT 5 UNION ALL SELECT 6 UNION ALL SELECT 7 UNION ALL SELECT 8 UNION ALL SELECT 9) AS b
        CROSS JOIN (SELECT 0 AS a UNION ALL SELECT 1 UNION ALL SELECT 2 UNION ALL SELECT 3 UNION ALL SELECT 4 UNION ALL SELECT 5 UNION ALL SELECT 6 UNION ALL SELECT 7 UNION ALL SELECT 8 UNION ALL SELECT 9) AS c
    ) a
    JOIN (SELECT @i := -1) r1
    WHERE 
    @i < DATEDIFF('2016-08-01', '2016-07-01')

) AS dateTable
WHERE WEEKDAY(dateTable.Day) IN (5,6)
ORDER BY dateTable.Day;

DEMO DE TRABAJO

Nota: WEEKDAY devuelve el índice de días de la semana para la fecha (0 =lunes, 1 =martes, … 6 =domingo).

EDITAR:

Si solo necesita count :

SELECT 
COUNT(*) AS total
FROM 
(   SELECT ADDDATE('2016-07-01', INTERVAL @i:[email protected]+1 DAY) AS DAY
    FROM (
        SELECT a.a
        FROM (SELECT 0 AS a UNION ALL SELECT 1 UNION ALL SELECT 2 UNION ALL SELECT 3 UNION ALL SELECT 4 UNION ALL SELECT 5 UNION ALL SELECT 6 UNION ALL SELECT 7 UNION ALL SELECT 8 UNION ALL SELECT 9) AS a
        CROSS JOIN (SELECT 0 AS a UNION ALL SELECT 1 UNION ALL SELECT 2 UNION ALL SELECT 3 UNION ALL SELECT 4 UNION ALL SELECT 5 UNION ALL SELECT 6 UNION ALL SELECT 7 UNION ALL SELECT 8 UNION ALL SELECT 9) AS b
        CROSS JOIN (SELECT 0 AS a UNION ALL SELECT 1 UNION ALL SELECT 2 UNION ALL SELECT 3 UNION ALL SELECT 4 UNION ALL SELECT 5 UNION ALL SELECT 6 UNION ALL SELECT 7 UNION ALL SELECT 8 UNION ALL SELECT 9) AS c
    ) a
    JOIN (SELECT @i := -1) r1
    WHERE 
    @i < DATEDIFF('2016-08-01', '2016-07-01')

) AS dateTable
WHERE WEEKDAY(dateTable.Day) IN (5,6)

Demostración