The boy won't go anywhere
There's a lot to talk about with regards to the ending of the game, but I'm going to leave most of that for the next page. Right now, we need to talk about stable time loops.
Now, I am a sucker for time travel in fiction. You throw in time travel and I'm all over it. FFVIII's way of doing it is, uh, convoluted, but seeing Squall get lost in time compression and land in the past was the moment that made me go holy shit this is so awesome. I can forgive all the ridiculous shit this game has put me through. (Except NORG.)
Stable time loops are, well, stable, and this one is pretty simple. Squall and Ultimecia go back to Squall's childhood, Edea receives Ultimecia's powers, Squall tells her about Garden and SeeD. Squall returns to the present, and in the far future, Ultimecia will eventually take control of Edea and set the entire game in motion again. Standard fare for a stable time loop. Squall gets to live out his happily ever after knowing that he defeated the sorceress in the future.
But here's the thing: what was the first iteration?
See, in every stable time loop, there has to be a first iteration that starts off the entire cycle. Because this is hard to understand without an example, we're going to use the Terminator movies.
(Why Terminator? Because 1: it meant I got to rewatch the movies in the name of site research, and 2: it's a franchise that's well-known enough that most people reading this will have at least a general idea of the premise. Also, we're only talking about the original film and T2 here, not T3 or Salvation, because T3 is godawful and Salvation is... yeah.)
Anyway, so the idea of Terminator is that if you kill the mother of your greatest enemy, he'll never be born. Likewise, if you destroy your enemy before it's ever built, the world is saved. However, both of these premises are dependent on the other, and there has to be a first iteration. The first Sarah Connor had a son named John, but his father wasn't Kyle Reese, because Skynet (and time travel) hadn't been invented yet. He was an entirely different person.
However, in that original timeline, Skynet is eventually created, takes over the world, and John Connor becomes enough of a problem that a Terminator is sent back to kill his mother. And Kyle Reese is sent to protect her.
What we see in the first film is not the second timeline. There are things in it that could not be part of the original timeline, but have become part of the loop — the photograph of Sarah, John's knowledge of his father. More details are added every time the loop resets, and there's no telling how many times it's been.
The problem about applying this to Final Fantasy VIII is that when you try to figure out what the original timeline could have contained, there's very little you can be absolutely certain about. Ellone exists, and Ultimecia wants to use her power to achieve time compression. Somehow, this is done, and as Ultimecia is about to die, she passes her powers onto Edea.
But is Squall involved? I really can't say. I can't even say that Rinoa is involved, or that she ever became a sorceress. Because in the original timeline, there is no Garden. There are no SeeDs. Squall tells Edea it was her idea in the game — she has no idea what he's talking about. Maybe she and Cid magically came up with the idea to create a school for mercenaries in the old Centra shelters out of the blue, but let's be real: they probably didn't.
The conclusion I can make about Final Fantasy VIII is that its plot is very much the result of extreme timeline littering. The game itself can't be the second iteration of the loop for SeeD to be so pervasive that it stretches all the way to Ultimecia's time. Like Terminator, I find it far more likely that every time the loop reset, more and more desperate measures had to be taken.
This comes up a lot in Terminator: The Sarah Connor Chronicles, in particular. While the series is nowhere near as good as the movies, it does have a lot of neat concepts, including the idea that every time they manage to defeat the next iteration of Skynet, all they do is push back the date it gets built. The timeline is so littered that it's going to be built one day, it's just a question of when.
The battle with Ultimecia is the same thing. She's going to exist one day. She's going to control Edea and try and achieve time compression. Defeating her doesn't change that eventuality, but it saves the future. And that's really all they can hope for.
(If you'd like to read more about time travel in media, I highly recommend checking out this site, which is where I first read about Terminator's original timeline. Highly recommended.)