The point of origin for a Fireball is the caster's finger, since you send a tiny bead to the detonation point.
There are, however, other spells with points of origin well away from the caster. You need a good line of effect, but AntiMagic doesn't block line of effect so it doesn't matter.
For example, Chain Lightning lets the caster pick the primary target, and the spell is centered there, after which it branches out to other targets within the limit of the area. As written, you could cast that from within an AntiMagic zone.
If you were standing near the edge, so the "corner of your square" was at the border, you could arguably cast ray, cone or line type spells. (You count range and area from that corner of your square, which is not within the Anti Magic zone.)
This is all "letter of the law" stuff, and as I said, it's arguable. I've argued it more than a little myself.
As a DM, you have to ask yourself, "Do I want to deal with this mess?". if the answer is no, then don't. Make a ruling that you can't cast while within, that you can't summon the magical forces in Anti Magic, no matter where they manifest.
Why is it a DM's call? (Aside from the fact that, in the end everything is, i mean?) Because the rules are unclear. They suggest that you might be able to, under some circumstances.
Your example of a 15 foot square with the monster standing in the middle is easy though: Under no interpretation could he project any spell that started at him (Fireball included), unless he had a 15x15 base (making "the corner of his square" coincide with the edge of the zone).
If you aren't the DM, then live with whatever the DM rules and move on from there.
In either case, whatever the DM rules is correct, at least for that game.