On 2b, Rashi "v'hichinu" says that according to Rabba, you need hazmana b'peh to allow use of an item on shabbos (or yom tov, as Rashi says later). And in explaining why an egg laid on a Sunday Yom Tov is assur, Rashi says because it needs to be muzmenes and muchenes from chol. So it's not just a problem that shabbos was meichin for yom tov, but that it didn't have hazmana from chol.
If so, why is an egg laid on a regular shabbos or yom tov mutar, (if not for the gzeirah of back-to-back shabbos and yomtov)-thre was no hazmana b'peh? And if you shecht a hen on a Sunday Yom tov and find eggs inside, they are mutar because there is no gzeira in this case-but how is it mutar, there was no hazmana b'peh on chol. Unless we have to make the ukimta that he actually was mazmin bpeh from chol, that if the egg is laid I'll eat it. But that sounds a bit dachuk.
Yaron Barach, Brooklyn , NY
(a) The example that Rashi gives in order to prove that Hazmanah b'Peh is necessary ("mi'Kan Ani Notel l'Machar...") is from Beitzah 10a, and 34b. In both cases, the matter under discussion is an item that is not yet designated for eating - i.e. Muktzah. Only in such cases would Hachanah b'Peh be necessary.
If an item is clearly designated for eating - such as eggs inside of a hen which are collected daily for eating - there would be an Umdena that it is Muchan for Shabbos (just like a hen that is being raised for slaughtering, in the Gemara above). No further Hachanah would be necessary.
(b) Regarding whether Rabah's Hachanah would have to be verbal ("b'Peh") - as we mentioned, Rashi's words are based on the Mishnah which specifies " Omer mi'Kan Ani Notel...". The word "Omer", however, can also mean "think" - see Gilyon Hashas to Zevachim 2a (especially the Rashi in Kidushin that he cites, which says it is not necessary to verbalize the Machshavah) and Tosfos Yom Tov to Uktzin 3:11 etc.