The narration goes as;

عن عائشة قالت لقد نزلت آية الرجم ورضاعة الكبير عشرا ولقد كان في صحيفة تحت سريري فلما مات رسول الله صلى الله عليه وسلم وتشاغلنا بموته دخل داجن فأكلها

Reported 'Aisha (RA): ‘the verse of stoning and of suckling an adult ten times was revealed, and they were (written) on a paper and kept under my pillow. When the Messenger of Allah (PBUH) expired and we were occupied by his death, a goat entered and ate away the paper.’ (Sunan Ibn Majah, Hadith 1944)

1- Authenticity of the narration:

Whenever we have a narration we ought to see whether it is authentic or not? The narration infact has some problems.

The particular chain given in Sunan Ibn Majah finds one of the narrators Muhammad bin Ishaq narrating it using the word عن ('an) which is rather an ambiguous way of narration and renders the narration weak when used by a narrator known for practicing Tadlis [practice of subtly missing a link] and Muhammad Ibn Ishaq is indeed such a narrator. Thus through particular chain of narration in Sunan Ibn Majah the narration is weak and unauthentic due the above mentioned defect though it has other issues as well as mentioned in the lines to follow. This is clarified by Shaykh Muhammad Taqi Usmani in Takmala Fath Al-Mulhim 1/69 pub. Darul Ahya Al-Turath Al-Arabi, Beirut.

In Musnad Ahmad the same narration is given through the same chain but with an explicit way of narration i.e. it does not have the defect like the narration in Ibn Majah’s collection. But the narration is exposed to more criticism because many other narrators have related from 'Aisha (RA) about the suckling/breastfeeding but no one has narrated the words found in this chain even though the narrators in those cases are more reliable and consistent than Muhammad bin Ishaq. And due to thefact of these words being narrated solely by him and in defiance to other much more reliable narrators, scholars have questioned its authenticity. Shaykh Shu’aib Arnaud has classified it as Da’if in his classification of Musnad Ahmad. See Musnad Ahmad 6/269 Hadith 26359.

2- The narration no way questions Qur’an infallibility:

Even if the narration were authentic it no way questions the claim of Qur’an being totally preserved and here are my evidences for this;

1- One of the two allegedly lost verses as per this narration was about stoning i.e. punishment of married adulterers. But other narrations prove that a commandment was revealed about stoning but the Holy Prophet (PBUH) did not allow it to be written as a part of the Qur’an implying that it was not meant to be Qur’an integral part. Following narrations testify to this;

a-It is reported in a narration from Kathir bin Salt that: Zaid (b. Thabit) said: 'I heard the Messenger of Allah say, 'When a married man or woman commit adultery stone them both (to death)', (hearing this) Amr said,

فقال عمرو : لما نزلت أتيت النبي صلى الله عليه وسلم فقلت : أكتبها ؟ فكأنه كره ذلك

'When this was revealed I came to Prophet and asked if I could write it, he (the Prophet) disliked it.' (Mustadrik Al-Hakim, Hadith 8184. Hakim called it Sahih)

b- About this 'verse' Kathir bin Salt says that he, Zaid bin Thabit and Marwan bin Hakam were discussing as to why it is not written in the Quranic manuscript and Umar bin Khattab was present with them and listening to their discussion he said he knew it better then them and told them that he came to Messenger of Allah and said:

يا رسول الله أكتبني آية الرجم قال فأتيته فذكرته قال فذكر آية الرجم قال فقال يا رسول الله أكتبني آية الرجم قال لا استطيع ذاك

"'O Messenger of Allah, let the verse about stoning be written for me.' He (the Prophet) said, 'I can't do this.'" (Sunan Al-Kubra Baihiqi 8/211 & Sunan Al-Kubra Nasai Hadith 7148. Albani (in Sahiha 6/412) said Baihiqi pointed to its authenticity)

Had it meant to be a part of the Qur’an why would Holy Prophet (PBUH) dislike its being written and who could stop him from doing it?

2- The second allegedly lost verse was about suckling of an adult ten times but in this case too we have other narrations which categorically say that the verse was abrogated. And interestingly those narrations come through 'Aisha (RA) only. In Sahih Muslim we read;

عَنْ عَائِشَةَ أَنَّهَا قَالَتْ كَانَ فِيمَا أُنْزِلَ مِنْ الْقُرْآنِ عَشْرُ رَضَعَاتٍ مَعْلُومَاتٍ يُحَرِّمْنَ ثُمَّ نُسِخْنَ بِخَمْسٍ مَعْلُومَاتٍ فَتُوُفِّيَ رَسُولُ اللَّهِ صَلَّى اللَّهُ عَلَيْهِ وَسَلَّمَ وَهُنَّ فِيمَا يُقْرَأُ مِنْ الْقُرْآنِ

'A'isha (Allah be pleased with, her) reported that it had been revealed in the Holy Qur'an that ten clear sucklings make the marriage unlawful, then it was abrogated (and substituted) by five sucklings ... (Sahih Muslim, Hadith 2634)

This narration explicitly says that verse about ten sucklings was abrogated. For further queries about this issue and the narration from Sahih Muslim see THIS.

Having known that neither of them was meant to be part of the Qur’an; even if we accept the narration in question we will have to say that perhaps 'Aisha (RA) had kept them with her as a historical record and nothing more. Thus even if the goat actually ate them up no part of the Qur’an was lost.

Moreover 'Aisha (RA) lived through the whole period of Qur’an compilation during the time of Abu Bakr (RA) and Usman (RA) while she was unanimously considered an authority for herself so if she had any thought about some verses missing she would have brought it to attention of other Companions of the Prophet (PBUH). Infact we have evidence of Usman (RA) making special endeavor of consulting 'Aisha (RA) and her records for verifying the official compilation. See Ibn Shabba’s Tarikh Al-Madina p.997. Despite all this she never raised the issue supporting our conclusion that no part of the Qur’an was lost even if the narration is considered reliable.

Indeed Allah Knows the best!