THE ANSWER OF R. YOSEF
Answer: The egg of the chicken designated for eating is prohibited as a Gezeirah with fallen fruit.
Question (Abaye): But fallen fruit is itself a Gezeirah (lest one climb up and cut a fruit), making the egg a Gezeirah l'Gezeirah!?
Answer: The egg is part of the same Gezeirah as Peiros ha'Nosherin.
THE ANSWER OF R. YITZCHAK
Answer: The egg is prohibited lest we permit juice which oozed from a fruit.
Question (Abaye): But that is a Gezeirah l'Gezeirah (Mashkin she'Zavu lest one come to squeeze the fruit)!?
Answer: The egg is included in the Gezeirah of Mashkin she'Zavu.
WHY THE ANSWERS DIFFER
The others differ from R. Nachman because the question regarding including both in the Mishnah was unresolved.
The others differ from Rabah because they do not subscribe to the Torah prohibition of Hachanah.
Question: Why doesn't R. Yosef subscribe to R. Yitzchak?
Answer: Because fruit and egg are both food, whereas juice does not fit in.
Question: Why doesn't R. Yitzchak subscribe to R. Yosef?
Answer: Egg and juice share their being secreted from their housing, unlike fruit which is always exposed.
By questioning the seemingly contradictory opinions of R. Yehudah, R. Yochanan demonstrates that he views the Gezeirah of Beitzah as similar to Mashkin she'Zavu (and they would be either prohibited or permitted together).
In one Mishnah, R. Yehudah permits the oozing juice if the fruit were intended for eating, thus he does not apply the Gezeirah of Mashkin she'Zavu and permits the extract of food (Uchla d'Efras).
In another Mishnah (dealing with implications of two days of Yom Tov), R. Yehudah prohibits an egg laid on Yom Tov from being eaten that day (thus seeing anything which oozes from its place, even food, as being Asur).
R. Yochanan resolves the difficulty by switching the positions of R. Yehudah and Rabanan in the first Mishnah (dealing with Sechitah).
By raising the ruling regarding Mashkin she'Zavu as a contradiction to the ruling regarding Beitzah, R. Yochanan has shown his view, as above.
Ravina resolves the apparent contradiction between the positions of R. Yehudah by asserting that R. Yehudah (in the second Mishnah) was only responding to the Rabanan given the latter's position.
R. Yehudah himself maintains that the egg would even be permitted on that day (Uchla d'Efras) thus his positions are consistent.
To the Rabanan who prohibit the egg, R. Yehudah asks that at least it should be permitted on the second day since the two days of Yom Tov are Shtei Kedushos.
The Rabanan, however, hold that they are one Kedushah.
Ravina bar R. Ula asserts that the contradiction does not begin since the chicken spoken of in the second Mishnah was designated for laying eggs and the issue in that Mishnah is R. Yehudah's position on Muktzah, not Mashkin she'Zavu.
Question: Rabah (who maintains the Torah prohibition of Hachanah) seems to be the only understanding of our Mishnah which also explains why a Safek egg is Asur and will not be Batel, among the Halachos cited in the Beraisa (whereas Mashkin she'Zavu and Peiros ha'Noshrin are Gezeiros d'Rabanan, and should be Mutar b'Safek)!?
Answer: This Beraisa is speaking of a Safek Tereifah.
Question: Then it should be Batel, unlike Safek Yom Tov which is a Davar she'Yesh lo Matirin (and would understandably not be Batel if it were d'Oraisa).
Answer: Beitzah is Chashuvah and is not Batel.
Question: That is only true according to the opinion which holds that anything which is, at times, counted is viewed as Chashuv, but will not hold according to the view (below) which only accords the status of Chashuv to that which is always counted!?
(R. Meir): Bundles of Tilsan of Kelai Kerem are not Batel (must all be burned).
(Chachamim): They are Batel in 201 (the normal Bitul for Kilayim and Orlah).
R. Meir views anything which [either, at times or always], is counted as Chashuv and not Batel while Chachamim restrict that status to six (or, according to R. Akiva, seven) items.
(R. Yochanan): The word is Es (always).
(Resh Lakish): The word is Kol (at times).
How, then, according to R. Yochanan, can an egg (which is not always counted) be Chashuv and not Batel?
Answer: This Tana (who discusses Litra Ketzi'os) maintains an extreme position, and accords the status of "countable and not Batel" even to that which is Asur d'Rabanan.