At the start of the book, we see the unnamed, curmudgeony hero sitting in a chair while a large, dog-like creature strolls by with Sam on his back, holding a sign that says "Sam-I-Am." Sam is depicted as more youthful and jolly, while our hero (who we'll just call Bill), seems old and forlorn. Sam isn't doing anything to Bill, but Bill is still annoyed by how much pride Sam has in his identity and declares how much he dislikes Sam because of it.
At this point, Sam launches a grappling gun of sorts with a plate of green eggs and ham on top of it right in front of Bill's face, beckoning Bill if he does, in fact, like the dish. On the surface, this could be seen as Sam being silly, but what Seuss is actually doing here is having Sam ask Bill if he is gay and likes penis, with the two eggs and slice of ham serving as a pretty clear phallic symbol. At this point, Seuss is making us clear of what his characters represent: Sam as a more stereotypical, flamboyant gay guy proud and at peace with himself, and Bill as someone in the closet who never felt comfortable with his urges because of his 1930s-1950s conservative upbringing (the book was released in 1960.) It's no coincidence either that Sam is portrayed with colors like red and yellow, while Bill is only black and white, reflecting his single-minded viewpoint of the world. Still, even though Bill is taller than Sam, the two share a similar physical structure and even wear similar hats, showing that even though Bill thinks they're opposites, they have much more in common than they think, that perhaps there is even a spectrum of sexuality and that people aren't defined or labeled by one particular thing.
But on with the book. When Sam asks Bill if he would like the green eggs and ham here or there, he does so with two large, red gloves with their index fingers extended, like the question is literally poking at something sensitive in Bill. But instead of providing a simple no to Sam's query, Bill gets offended and provides a lengthy "I'll never eat them here or there, anywhere, or any time, Sam-I-Am." This is supposed to represent Sam asking Bill if he has ever had feelings in a non-heterosexual way, and Bill getting defensive and going on some sort of homophobic diatribe instead of just saying no, I don't swing that way.
Sam then asks Bill if he'd like green eggs and ham in a house and with a mouse, literally inside the house as he's asking. This represents Sam explaining to Bill how non-hetero people aren't heathens, and still live normal lives and do normal things. Of course, Bill provides his lengthy diatribe once again. Sam asks how about with a fox in a box? Obviously an unconventional situation that no one would enjoy, but now Sam is kind of poking fun at Bill's comments, kind of playing into Bill's mindset that all gays are just crazy people who do nothing but hang out and do drugs/hang out with foxes in boxes all day.
Now Sam himself is a bit annoyed and becomes more direct and aggressive, literally driving a car into Bill and onto a train. At this point, he's not even asking Bill if he wants green eggs and ham, but now commanding Bill to eat them, even telling Bill he will probably like the dish. We can interpret this as Sam now being more aggressive and starting to flirt with Bill, saying "hey, I'm going to put my hand on your knee/kiss you on the cheek, and if you don't feel anything, then you're straight." But Bill still relents. Throughout this whole passage, it's no coincidence at all that this is happening on a train heading into a tight, dark tunnel, which I don't have to tell you what that represents.
Sam asks Bill if he could eat them with a goat, an animal with horns, essentially the same thing of if he'd ever be willing to do a devil's threesome. Sam asks if Bill would like them on a boat, which represents him asking Bill would he ever be with a man if it was say on a foreign country or in the middle of the ocean and no one on the mainland knew about it.
But Bill finally relents, asking if he'll try the green eggs and ham, if Sam will leave him alone. This is essentially the same thing as Bill agreeing to let Sam kiss him or do more explicit stuff, if he'll just drop the topic. But of course, Bill loves green eggs and ham and proclaims all the places he'll eat them, thanking Sam-I-Am. Essentially, Bill has come out of the closet at this point, or at least admitted that he does have urges and they aren't bad and he can still live his life the same way.
But we can't ignore how Sam's line of questioning was needlessly invasive, after Bill quite clearly said he didn't want to try green eggs and ham from the get go. While Bill is happy now that he's open with his sexuality, Sam's constant prodding reflects the harm of outing in the LGBT community, where one is pushed by outside forces to reveal themselves causing great personal and societal damage, instead of letting that person do so on their own terms. While Bill is happy with his new sense of self, Seuss here is asking us if it was worth it since Sam was so reckless and hazardous in his needless questioning.