1. An egg laid on Shabbos:
An egg which has been laid on Shabbos is forbidden to be eaten and is therefore Muktzah. It is even forbidden to merely touch the egg. Nevertheless it is permitted to place a vessel over the egg to protect it. However it is forbidden for the vessel to touch the egg in the process.
If the egg became mixed with other eggs that were laid before Shabbos then all the eggs are forbidden to be eaten and they are all Muktzah. This applies even if the ratio of the mixture is 1:1000.
If an egg which was laid on Shabbos and the next day is Yom Tov or vice versa, then the egg remains Muktzah also the next day.
All the above applies even if there is a doubt as to when the egg was laid, in which case we are nevertheless stringent to forbid eating the egg and its entire mixture.
2. Fruits which fell off a tree:
All fruits which have fallen off a tree on Shabbos are forbidden to be eaten until after Shabbos and are therefore Muktzah. [Furthermore, even if there is doubt as to if the fruits fell on Shabbos or beforehand it is forbidden.]
Are vegetables which have become detached from the ground on Shabbos forbidden?
Yes. Hence they are Muktzah.
Tikkun Keli by foods:
Using straw or hay as a tooth pick: Therefore it is permitted to cut even with a knife [even into a particular measurement] straw or hay and [use it] to pick at his teeth [i.e. to use as a toothpick].
Using a twig as a tooth pick: However a twig which is not animal food, even to take it in order to pick at his teeth is forbidden [due to it being Muktzah, and to cut a piece off is forbidden also due to the fixing a vessel prohibition].
 Michaber 322/1; Admur 513/1-3
 The reason for this is because an egg which has been laid on a Shabbos which followed a Yom Tov [i.e. Yom Tov was on Friday], or on a Yom Tov which followed Shabbos [i.e. Yom Tov was on Sunday] then that egg is Biblically forbidden to be eaten. The reason for this is because the egg was already prepared to be laid the day before it was actually laid. Thus it ends up that Yom Tov or Shabbos prepared the laying of the egg of Shabbos/Yom Tov, and the Torah states “And on the 6th day prepare [for the Sabbath]”. From here it learned that only a weekday may prepare for Shabbos while a Holiday may not prepare for Shabbos. Likewise Shabbos may not prepare for Yom Tov, as Yom Tov is also called Shabbos. [513/1] Now, although Biblically an egg which was laid on Shabbos that did not follow a Yom Tov is permitted, nevertheless the Sages made a decree against eating any egg laid on Shabbos or Yom Tov, as a safeguard around the Biblical prohibition. [513/2]
An egg laid on Motzei Shabbos: May be eaten as only an egg which is being eaten on Shabbos or Yom Tov is forbidden to be prepared on Shabbos or Yom Tov. [513/1]
Other preparations done on Shabbos: Only matters which are prepared from Heaven are forbidden when done on a Shabbos before Yom Tov, however preparations done through the hands of man are not Biblically forbidden. [ibid]
 Since the egg is forbidden to be eaten it serves no purpose and is therefore Muktzah. [513/3]
 513/3; Now although all Muktzah is allowed to be touched if one does not shake it in the process, nevertheless by an egg it is forbidden as an egg is oval and even mere touching will cause it to move in the process. [ibid]
 As the egg laid on Shabbos is a forbidden food which will eventually become permitted, of which the ruling is that it is never nullified even in 1000x. [513/4]
 Michaber 322/2
This is due to a Rabbinical decree, as it ends up that the day before prepared the egg, as it is better to eat an egg which has been out of the chicken for some while then an egg which has just emerged. Nevertheless this is only a Rabbinical prohibition, as Biblically the egg was already prepared before Shabbos, and it is only when it was prepared to be hatched on Shabbos that it is forbidden the next day. [513/9]
 Michaber 322/3
 The reason for this is due to a decree one may come to pluck a fruit from the tree on Shabbos. Alternatively it is because the fruit is Muktzah as one did not have it in mind from before Shabbos. [M”B 322/7]
 Ketzos Hashulchan 146/23 from Peri Megadim, Upashut.
 M”B 322/5; brought in Ketzos Hashulchan 146/23
 M”B 322/6 in name of Peri Megadim
 M”B 9
 M”B 10
 In which case it would normally contain the cutting prohibition, although here it is allowed being that it is a food, and by foods the cutting prohibition is not applicable. [M”B 12]
 If done with one’s hand it is Rabbinically forbidden, and if done with a vessel is Biblically forbidden. [M”B 13]