There is a hadith that people who break the nafilah fast should perform qaza fast:
Aisha (r.a.) said: “We began fasting voluntarily one morning and then food was given to them and they broke their fast with it. Then the Messenger of Allah (saws) came in. Hafsa asked: ‘O Messenger of Allah (saws), A’isha and I began the morning fasting voluntarily and then food was given us and we broke the fast with it.’
The Messenger of Allah (saws) said: ‘You have not sinned, but fast another day in its place.’ ” (Tirmidhi, Siyam,36; ;Abu Dawood, Sawm,73)
Imam Al-Adham and Imam Malik say that qaza of breaking nafilah fast is wajib, based on this hadith. This hadith is weak, though. Because there is Zumayl amongst the narrators, who is criticized. However, this hadith was narrated by Ibn Hibban, Ibn Abi Shayba and Taberani through other sanads (chain of narrators).
Scholars who share this idea also base on the verses meaning “…do not invalidate your deeds.” (Mohammad 47:33) and “…complete your fast till night-fall” (Al-Baqarah/ The Cow 2:187). Because there isn’t discrimination of fast being fard or nafilah as completion of fast is commanded.
Following information is in the “Buyuk Islam Ilmihali” by Omer Nasuhi Bilmen:
“There is not a reason in religion for nafilah fasts to be performed. These are fasts for only people who voluntarily intend to have good deeds. However breaking it, needs qaza. The reason for this qaza is that the worship was began for the sake of pleasing Allah. Since interrupting it would be non-permissible, completing it would be wajib.” (Ömer Nasuhi Bilmen, Büyük İslam İlmihali, İstanbul, 1986, p. 255-256.)
Shafi and Hanbalis have the opinion that breaking nafilah fast does not require qaza.