It is mustahabb to sacrifice the ‘aqeeqah on behalf of the new born on the seventh day, because the Prophet (blessings and peace of Allah be upon him) said: “Every child is in pledge for his ‘aqeeqah, which should be sacrificed on his behalf on the seventh day, and his head should be saved and he should be given his name.” Narrated by Abu Dawood, 2455; classed as saheeh by Shaykh al-Albaani. 

Ibn Qudaamah (may Allah have mercy on him) said: Our companions said: The Sunnah is to sacrifice it on the seventh day, and we do not know of any difference of opinion among the scholars who say that it is prescribed concerning the fact that it is mustahabb to sacrifice it on the seventh day. The evidence for that is the hadeeth of Samurah, from the Prophet (blessings and peace of Allah be upon him) according to which he said: “Every child is in pledge for his ‘aqeeqah, which should be sacrificed on his behalf on the seventh day…”

End quote from al-Mughni, 9/364 

Secondly: 

Once it is established that it is mustahabb to sacrifice the ‘aqeeqah on behalf of the child on the seventh day, is the day of birth included in that, according to the majority of scholars? An-Nawawi (may Allah have mercy on him) said: Is the day of birth counted as one of the seven? There are two opinions, the more correct of which is that it is counted, so the sacrifice is done on the sixth day after (the day of birth). 

The second view is that it is not counted, so the sacrifice is done on the seventh day after (the day of birth). This is what is stated in al-Buwayti, but the first view is the apparent meaning of the hadeeths. If the child is born at night, the day that follows that night is included, and there is no difference of opinion on this point. 

End quote from al-Majmoo‘, 8/411 

In al-Mawsoo‘ah al-Fiqhiyyah (30/279) it says: The majority of fuqaha’ are of the view that the day of birth is counted as one of the seven, but the night is not counted if the infant is born at night; rather the day that follows that night is counted. End quote. 

Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) said: The words “is to be sacrificed on the seventh day” mean that it is Sunnah to sacrifice the ‘aqeeqah on the seventh day. So if the child was born on Saturday, the sacrifice is to be done on Friday, i.e., one day before the day on which the child was born. This is the basic guideline. If the child was born on Thursday, then (the ‘aqeeqah) should be on Wednesday.

End quote from ash-Sharh al-Mumti‘, 7/493 

Thirdly: 

What was said about the day of birth not being counted if the child is born after midday was stated by a number of scholars (may Allah have mercy on them). In fact they said that the day of birth should not be counted at all, whether the child was born before or after midday. This is the view of the Maalikis. 

It says in Mukhtasar Khaleel: It is recommended to sacrifice one sheep that meets the requirements of udhiyah on the seventh day after birth, during the day, and the (previous) day should not be counted if the birth occurred shortly before Fajr. 

Al-Mawwaaq (may Allah have mercy on him) said, quoting from Ibn Rushd: The view of Ibn al-Qaasim and his report from Maalik in al-Mudawwanah and elsewhere is that if the child was born after dawn, that day is not to be counted, and the seven days should be counted from the following day. If the child is born before dawn, if that was at night, then that day is to be counted.

End quote from at-Taaj wa’l-Ikleel, 4/390 

The correct view is that of the majority of scholars (may Allah have mercy on him), which is that the ‘aqeeqah is to be sacrificed on the child’s behalf on the seventh day from his birth, because the Prophet (blessings and peace of Allah be upon him) said: “[it] should be sacrificed on his behalf on the seventh day…”. 

Shaykh Muhammad ibn Muhammad al-Mukhtaar ash-Shanqeeti (may Allah preserve him) said: What is meant is that the day of the ‘aqeeqah is the seventh day after the day of birth. Therefore the day (of the week) on which he is born is the seventh day.

End quote from Sharh al-Mustaqni‘. 

The matter is mustahabb. If it is possible to sacrifice the ‘aqeeqah on the seventh day after his birth, this is better, but if it is not possible to do it until after the seventh day, there is nothing wrong with that and the ‘aqeeqah sacrificed on the child’s behalf will be acceptable. 

An-Nawawi (may Allah have mercy on him) said: If he sacrifices it after the seventh day or before it, or after the birth, that is acceptable, but if he sacrifices it before the birth, that is not acceptable, and there is no difference of scholarly opinion on this point; rather it is just meat.

End quote from al-Majmoo‘, 8/411 

And Allah knows best.