A person is wearing a Ring of Counterspells which is storing a Fireball spell. When this person is in the AOE of a fireball is the entire fireball countered? Do they negate it just for their square? Is there some other outcome?
A ring of counterspells only triggers when the wearer is the target of the same spell it stores. Since area-effect spells target a location, not a creature, the ring can't be triggered by merely being in the AoE of a fireball. Being merely caught in the blast area isn't enough to activate the spell-countering effect.
The ring might counter a fireball that directly impacted you, at the GM's discretion. Fireball is a bit of an odd-ball spell for an area-effect spell, since it can "impact" something between the caster and the target location:
[if the glowing, pea-sized bead] impacts upon a material body or solid barrier prior to attaining the prescribed range, [it] blossoms into the fireball at that point
A GM could rule that this is enough interaction between the wearer of the ring and the fireball to count as having the spell being cast directly "upon" the wearer – even though fireball doesn't have a creature target – that the ring would trigger.
This is in the official 3.5 FAQ, but honestly the answer is somewhat hard to generalize from. Pathfinder does rule differently on some issues, of course.
Can a ring of counterspells counter a fireball spell or other area effect if the wearer is in the affected area?
The ring of counterspells counters spells that are cast on you only, not necessarily a spell that affects you. So if you were the target of a fireball, then the ring would counter it; however, if you were just standing too close to the intended target, you would be affected normally and the ring of counterspells would not come into play.
I would personally interpret the ring as working the same way a readied action to counterspell would work -- that anytime you'd be affected by the spell, the ring would counterspell it for you. This is not 100% clear from the rules, and is specifically contrary to the FAQ answer. (Which was 3.5 in any case.)