As Magician mentioned, we know almost nothing about the Green and the White.
Well, we know a little. The Green is imprisoned by the Elf Queen (18, 26), and is associated with poison.
The White was killed by the Lich King, and turned into an undead (26). It is associated with cold. There are still cults and religions devoted to the White (273), but are more in denial than looking at some current, glorious, figure.
The Green is by far the most flexible of these. After all, it's still alive, and I'm assuming that's a huge step on the path to power in the 13th Age (undead are powerful, but with the Lich King I'd doubt that even a dragon could break free that easily). Were it to rise to power it'd likely hold a huge grudge against the Elf Queen or be plainly and purely evil. Alternatively, as it is a thrall of the Elf Queen, it could rise as a puppet Icon or perhaps be used in a war against the Lich King.
We don't know much about Green's relationships. His reappearance would be a little awkward, I'd imagine, at least for certain parties. The Great Gold Wyrm may or may not care; so long as he isn't attacked and his followers are not harassed he's usually pretty easy going and occupied by the whole "I'm stuck keeping demons from killing everyone and eating their spleens" thing. For The Three, they don't seem like the "Welcome back to the group!" kind of dragons, so I'm betting there would be a new rivalry, especially if the Green and White returned to prominence as Icons around the same time. At the same time, I doubt that the Green would even care to return to The Three; green dragons tend not to get along with the rest of dragonkind, and are very arrogant about it. This probably played a role in his capture in the first place.
The White is dead, but it's implied to have been resurrected. We don't know how much free will that something on the scale of a reanimated dragon has; lower-level undead have little, while something like a true vampire has more. It's realistic to say that it's likely to make an appearance, but not be on the same level of former glory. It will likely appear as a villain, either of its own accord or operating in allegiance with the Lich King.
Were the White to return independently, he would likely try to recover his former glory; he was almost certainly an Icon, or at least a lower-case-I-icon, so there's no reason he couldn't lead his own faction. Depending on the mood and time, he could be evil and corrupted, or something more like the Crusader, or even potentially reject the Lich King's ways and become a "good" character, but that's all speculative. Either way, if the White returns, he's already got a framework to build on.