Fasting 6 days of Shawwal is highly recommended by the Sunnah. However, these 6 days are not the white days, which occur in every month.
The late Sheikh `Atiyyah Saqr, former head of Al-Azhar Fatwa Committee, states: The white days do exist in every lunar month; such days occur when the moon appears all the night; hence the name relates to the appearance of moon all the night and the sun all the day; they include the 13th , the 14th , and the 15th days of every lunar month. (So, they are 3 days, not six as common).
It is also said that they are given this name because Almighty Allah granted Adam forgiveness and whitened his Book of Deeds (purged it of sins) during these days. It has also been reported that when Adam came down to earth, his skin became black and thus Almighty Allah commanded him to fast for these days. When he finished the first day of fasting one third of his skin became white and by the lapse of the third day, his body became totally white. However, this narration is known to be incorrect and falsified.
Anyhow, Islam makes it recommended for Muslims to fast for these three days. Ibn `Abbas is quoted as saying: “The Prophet (peace and blessings be upon him) was of the habit of fasting the white days while resident or on journey.” Hafsah is also reported as having said: “There are four things the Prophet (peace and blessings be upon him) never abandoned; fasting the day of `Ashoura, the first ten days of Dhul-Hijja, three days of every month and the two rak`as of Dhuha.” (Reported by Ahmad)
Az-Zurqanisaid: “The wisdom behind these three days is that they fall in the middle of the month and the lunar eclipse most likely occurs during them. Once this occurs, there will be great reward for the person, for in such a case he will be fasting and performing many other acts of worship. This is as regard the known 3 days that are recommended for fasting.
Now to the six days of the month of Shawwal. These days cannot be called white days. In addition, it is merely recommended, and not obligatory, to spend these days in fasting. It is reported that the Prophet, peace and blessings be upon him, said: “He who fasts the month of Ramadan and follows it with six days of Shawwal, will be rewarded as if fasting for the whole year.” (Reported by Muslim)
It is also allowable for any Muslim who has to make up for missed days of Ramadan to have two intentions; one for making up for the days he/she missed and the other for fasting the six days of Shawwal. However, if he made the two types of fasting in separate days, this will be better. Based on this, it is permitted for a person who finds it more difficult to make up for the missed days of Ramadan and fast the six days to have two intentions for both acts and perform them in the same number of days. He can also intend making up for the missed days and later perform the Sunnah fast for six days.