Gemara

2a - Mishnah - Beis Shamai: One may eat an egg laid on Yom Tov;

Beis Hillel forbid eating it.

Gemara - Question: What is the case?

If the owner is ready to eat the hen (i.e. he is not saving it for eggs), why would Beis Hillel forbid? The egg is like part of a food that separated off!

If he keeps the hen for eggs, why would Beis Shamai permit? The egg is Muktzah!

Answer #1 (Rav Nachman): Really, the hen is for eggs. Beis Hillel hold like R. Yehudah, who forbids Muktzah and also Nolad. Beis Shamai hold like R. Shimon; who permits Muktzah and Nolad.

2b - Objection: If they argue about Muktzah, why do they argue about the egg? They should argue about the hen itself, to teach that Beis Hillel forbids even Muktzah (that is not Nolad)!

Suggestion: They argue about the egg to teach the extremity of the lenient opinion.

Rejection: If so, they should argue about both of them (so we will also understand Beis Hillel)!

Answer #2 (to question 3 - Rabah): Really, the owner is ready to eat the hen. The case is, Yom Tov is on Motza'ei Shabbos. Beis Hillel forbid because of Hachanah (preparation).

Rabah holds that an egg is finished (ready to be laid) one day before it is laid.

This is like Rabah taught elsewhere:

Rabah: "V'Hayah ba'Yom ha'Shishi v'Hechinu" - a weekday can prepare for Shabbos or Yom Tov, but Yom Tov cannot prepare for Shabbos or vice-versa.

Question (Abaye): If so, if Yom Tov is not on Motza'ei Shabbos the egg should be permitted!

Answer: Chachamim decree to forbid lest people permit when Yom Tov is on Motza'ei Shabbos.

Question (Abaye): If so, an egg laid on Shabbos should be permitted!

Answer: Chachamim decree lest people permit when Shabbos is on Motza'ei Yom Tov.

Answer #3 (Rav Yosef): Chachamim forbid the egg lest people permit fruit that falls off trees on Shabbos (these are similar).

Answer #4 (R. Yitzchak): Chachamim forbid the egg lest people permit juice that flows from fruits on Shabbos.

3b - Beraisa: An egg laid on Shabbos or Yom Tov is forbidden. One may not even move it. A Safek about such an egg is forbidden. If it became mixed with 1000 eggs, are all forbidden.

Rishonim

Rif: We establish the Mishnah like Rabah. A Beraisa forbids even to move an egg laid on Shabbos or Yom Tov. A Safek about such an egg is forbidden. If it became mixed with 1000 eggs, are all forbidden.

Rambam (Hilchos Yom Tov 1:19): A weekday can prepare for Shabbos or Yom Tov, but Yom Tov cannot prepare for Shabbos or vice-versa. Therefore, an egg laid on Yom Tov that is Motza'ei Shabbos is forbidden, even if the owner is ready to eat the hen. The egg was finished the day before it is laid, so Shabbos prepared for Yom Tov. Chachamim forbid every Yom Tov lest people permit when Yom Tov is on Motza'ei Shabbos. Likewise, every Shabbos is forbidden lest people permit when Shabbos is on Motza'ei Yom Tov.

Rambam (20): Just like one may not eat the egg, one may not move it. Even if it became mixed with 1000 eggs, all are forbidden

Magid Mishnah: The Rambam did not need to say that a Safek is forbidden, for he said that it cannot be Batel.

Rosh (1:1): The Rif rules like Rabah, who holds that Hachanah is mid'Oraisa. Therefore, we are stringent about a Safek.

Poskim

Shulchan Aruch (OC 322:1): If an egg was laid on Shabbos it is forbidden even to move it. Even if it became mixed with 1000 eggs, all are forbidden.

Shulchan Aruch (513:1): If an egg was laid on Yom Tov it is forbidden to touch (Rema - i.e. move) it. Even if it became mixed with 1000 eggs, all are forbidden.

Magen Avraham (1): Even if one was ready to eat the hen, we forbid the egg lest one eat an egg laid on Yom Tov after Shabbos, which is forbidden due to Hachanah.

Shulchan Aruch (2): If we are unsure whether it was laid on Yom Tov or Chol, we are stringent.

Magen Avraham (4): The Tur forbids a Safek, and even if it (a Safek) became mixed with 1000. The Shulchan Aruch reversed the order and forbids even if it became mixed with 1000, and forbids a Safek. This suggests that the Shulchan Aruch permits if a Safek became mixed. Even though in this very case the Shulchan Aruch (YD 102:1) forbids a Safek that became mixed, perhaps that refers to a Safek mid'Oraisa, i.e. it was born on Yom Tov after Shabbos. Nevertheless, one should be stringent, for many forbid a Sefek-Sefeka when Yesh Lo Matirim.

Mishnah Berurah (5): Some say that we may be lenient because it is a Safek mid'Rabanan, and even if the Safek became mixed with only two others. To avoid a loss one may be lenient.

Beis Yosef (DH Kosav ha'Ran): Some permit to buy eggs from Nochrim even on the first day of Rosh Hashanah, since most eggs were not laid today. This is the Nochri comes to sell to the Yisrael. It is forbidden to buy from the Nochri's house, for perhaps he sells eggs laid that day and the Isur is Kavu'a, in which case we may not follow the majority. Perhaps in every case we may not follow the majority, because Yesh Lo Matirim. Or, perhaps that is only when we know that there was an Isur (but here, perhaps all the eggs are permitted).

Shulchan Aruch (513:5): If an egg was laid on Yom Tov Rishon it is permitted on Yom Tov Sheni of Chutz la'Aretz, but not on Yom Tov Sheni of Rosh Hashanah.

Beis Yosef (ibid.): Ra'avan says that the Ge'onim decreed to forbid the egg on every Yom Tov Sheni, lest people permit on the second day of Rosh Hashanah. This is unlike all my Rebbi'im.

Shulchan Aruch (ibid.): Also, if Yom Tov is adjacent to Shabbos, an egg laid on one is forbidden on the other

Magen Avraham (8): The Yam Shel Shlomo says that according to one answer of Tosfos, a birth is called Hachanah, so it is forbidden mid'Oraisa on the second day. This is wrong. Milchamos Hash-m says that (mid'Oraisa) since the first day was Mechin for itself, it is permitted the next day. Mid'Rabanan, the egg is forbidden on the first day, so that day cannot prepare for the second.

Rema: If Yom Tov Rishon is on Motza'ei Shabbos, an egg laid on Shabbos is permitted on Yom Tov Sheni.