Trivia: The film had a notoriously troubled production, with several writers and directors dropping out of the project during development. With a looming release date, sets were built and a crew was assembled before a script was even settled on, and the final draft had to be written around the sets that had already been built. David Fincher, then a popular music video and commercial director, was chosen to helm the film, but he had nonstop creative difficulties with the producers and studios. He has since gone on to disown the film, as he feels it isn't reflective of his vision.
Trivia: Originally, the movie was going to be shot by cinematographer Jordan Cronenweth, but Cronenweth suffered Parkinson's Disease midway through filming and was let go by the line producer, being replaced by Alex Thomson. However, impressed by Cronenweth's work, David Fincher later worked with Cronenweth's son Jeff in nearly all of his future movies, resulting in two Academy Award nominations.
Trivia: The plot of the movie is a combination of two spec scripts written for the film - one was by David Twohy (Riddick) who wrote a script set on a prison planet, and another by Vincent Ward, set on a monastery planet with the same characters and basic plot. Despite this, neither received credit for the screenplay (Although Ward received story credit).
Answer: Agreed the Queen could have stowed the eggs away on her body as Alien bodies, especially one her size, could easily have a couple of eggs hidden upon her, and we would not know. They are masters of disguise these creatures and can merge in with their own environments or others...what's to say an egg can't look like part of the Queen's body? In addition, could Facehuggers not attach themselves to the Queen's body and merge with it somehow? Flatten themselves against her skin maybe?