It is a common practice for people to buy their Pesach eggs in advance of Pesach. I've seen different reasons for this, including:
The popular notion that some chametz might somehow get into the egg, and therefore the egg must be owned by a Jew in advance of Pesach when the chametz in the egg is nullified (batel b'shishim). A possible means for chametz getting absorbed into the egg is discussed and dismissed by R' Moshe Feinstein, although he recommends requiring people to clean the outside of their eggs before cooking (Igros Moshe OC vol. 3, § 61).
The concern of benefiting from an animal derivative whose creation is partially due to chametz that the animal may have eaten during Pesach, as chicken feed typically includes some of the five types of grain that can become chametz.
If an egg is stamped, the ink sometimes includes grain-derived ethanol. To quote the Star-K website: "Most authorities are of the opinion that alcoholic beverages such as whiskey, which is derived from wheat, barley or rye, are chometz gamur, and a person must not own these products on Pesach (Mishna Berurah Orach Chaim 442 Shaarei Teshuva ibid. )." According to the London Beth Din, such eggs can be permitted on the grounds that any trace of ethanol in the ink evaporates immediately after stamping.
The second point above is discussed by acharonim, mostly with respect to milk. A major point of debate is whether zeh vazeh goreim (something whose existence results from both prohibited and permissible causes) is permissible when one of the causes is forbidden chametz on Pesach. For example, the Magen Avraham (OC 445:5) rules it is forbidden, while the Nishmas Adam (vols. 2-3, § 119:9) permits it (and in fact permits drinking such milk on several grounds1 and implies that zeh vazeh goreim exists even if an animal's diet is exclusively chametz2). In the case of eggs, commercial chicken feed generally consists of both grain and non-grain ingredients, so presumably there are two gormim present.
There is additional reason to be lenient with the milk of animals owned by non-Jews, as non-Jews are not subject to the prohibition against owning or benefiting from chametz on Pesach (see Beis Efrayim OC, end of responsum 35).
While there seem to be strong halachic similarities between eggs and milk, I found only limited explicit discussion among the acharonim regarding eggs purchased on Pesach.
Assume a Jew purchases eggs on Passover that were from a farm owned by non-Jews. Are there contemporary poskim that rule on whether a Jew may eat the eggs, and, if so, what do they say? Does it make a difference if it is sha'as had'chak (e.g. a person would have a very limited diet without those eggs, and his health may be affected by abstaining)? Does it make a difference whether the eggs are purchased during the first half of Pesach (or so), when they were most likely laid before the holiday?
1 One ground for leniency presented in the Nishmas Adam appears to be a novel ruling that milk from a kosher animal cannot technically become forbidden even via a single goreim due to the general g'zerias hakasuv permitting milk of kosher animals. If so, this reasoning might be extended to eggs based on the Behag (§ 67, see also § 63) and Tosafos (Chullin 64a, second s.v. she'im rikmah) who appear to say that a similar g'zeiras hakasuv exists for eggs.
2 He considers the animal itself to be one of the gormim. R' Moshe Feinstein (Igros Moshe OC vol. 1, § 147), in a complex p'sak that does not entirely rely on zeh vazeh goreim, permits drinking milk that was produced on Pesach even if the animal's diet consists solely of chametz. Rather than assume that the animal is a goreim, R' Moshe rules that even echad goreim is permissible in the case described.