It's an age old question asked ever since Alien 3 hit theaters back in 1992. I've read many fan theories, attempts at explanation, but each and one of them created more discrepancies and holes. There were different ways explaining the reappearance of Alien in the older scripts of the film, which dealt with a different story, but they don't answer where did the egg came from on Sulaco in the actual story that was filmed, however one idea slightly points in the actual direction, but we'll get to that later.
So throughout the last 20 plus years, when internet forums became active and popular, I've read all kinds of fan theories, ranging from Bishop being an actual villain who smuggled the egg to the Queen carrying an egg on her back. Well, Bishop was never intended to be a villain, just the opposite. The trick was, he was suppose to be a shady character that we believed will eventually turn out to be like Ash, but the twist was that he wasn't. He was the first of Jim Cameron's (shortlived) storytelling tradition - in both Aliens and The Abyss, a character with a religious name appears to be a bad guy but turns good (Bishop in Aliens, Monk in The Abyss). So that's just fan theory going against the vision of the creator. As far as the egg on the Queen's back, that's a silly theory that doesn't really need much debunking, so I'll omit about a hundred reasons as to why that's not possible, but I'll just mention we do see her backs. Many think she wouldn't be able to lay an egg because she ripped away from her ovipositor, however ovipositor's function is only to transfer the eggs safely, it isn't an actual reproductive system. So she could, but at the same time she couldn't.
See, the problem isn't the Queen lying an egg, the actual problem is how the Sulaco survivors could actually miss it, especially since in Alien 3 we find out that Sulaco could detect movement AND an actual alien life form on board! So what the hell?
Not only that, but I don't think I need to mention the obvious thing - the ship would've been checked inch by inch before going to cryo, especially knowing that a 14 foot tall Queen could stow away, nevermind the smaller iterations. Ripley would never go to sleep without making absolute sure. And the area covered was very small - just the dropship, because in the hangar bay, the Queen was being watched by Bishop and Newt the entire time since she exited the dropship. AND they had ship who could do it instantly.
So the Queen couldn't bring, lay or leave an egg there. So was it just one of those cheap "deal with it, it makes no sense and we won't attempt to explain it, we needed a sequel" deals? Not exactly. While of course, Alien 3 has a fair share of issues, and the movie doesn't really explain it well at all, there are clues as to how did the egg got there in the first place. And how a facehugger implanted 2 embryos while the first movie clearly establishes that the creature lives only to plant a seed and dies.
So the Queen actually did not bring, lay or hide an egg. And Ripley and the crew and the Sulaco detectors could check the ship inch by inch, and nothing would have been found, because nothing was there before they went into cryo sleep. So wait, what?
Well, a tiny clue is in one of the earlier proposed Alien 3 stories/scripts, where an egg grows out of Bishop. No, no, I'm not saying that is what happened in the actual story they went with because it clearly didn't (and it's stupid), but an idea is introduced - Bishop grew an egg in earlier stories from the area where the Queen impaled him, so she was able to leave some residue, enzymes, that created an egg. And while an egg does not grow from Bishop, her "royal jelly" drips on and below the deck and eventually grows into egg while the crew is in the hypersleep. Where did I get that from? Is it a theory? No. See, first of all, David Fincher wanted the Alien to be even more insectile and wanted him to be like a spider - something confirmed by a puppeteer in the Cinefex magazine. Spider eggs often have the appearance of a drop of fluid. But that's of course, very little to back that statement, but I thought I'd mention it as a supporting argument. But the main thing is actually a photo of the Egg prop from Alien 3 in the Japanese 1992 Making of Alien 3 book. To my surprise when I first saw it, it wasn't just one egg. It was multiple eggs, of which two full size eggs actually see a bit of in the film, see below..
...and another, growing egg! Clearly growing from the residue it's in. See below
Clearly the Eggs did not originate from placement in the traditional sense, but instead grew from the dripping webbing. Below, photo of Alien 3 Full size egg prop and the new, growing egg from the "jelly"
So the Queen leaked its yolk and royal jelly which grew into almost 6 eggs in the course of days. The presence of 2 facehuggers is confirmed in the Alien 3 Novelization. One is trying to get to Newt's cryotube, the other one impregnates Ripley. As to why the eggs open without a presence of a host next to it, well, there isn't really an explanation for that one. The sequel had to happen, and many rules established had been broken. The Facehuggers themselves are a separate story for another time
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