Grappled (one hand free) or pinned (no hands free) seems to be the best fit.
The argument came about from a monster attempting to get out of a collapse I caused using a breath weapon. Since there is no condition and it does not specify that they are suffocating, just that they take non-lethal damage from an undefined source, the DM allowed the monster to attempt to escape via BW. Instead of the regular STR check.
Since the damage doesn't use the same rules as suffocation, I'd assume it's not strictly suffocation.
The rules-as-written here are very vague. There seems to be no explicit restriction on a characters actions. Although it should be obvious that movement and other such actions should be prohibited. Although I would hope that casting teleport is a viable option. This is one of those areas that rule 0 is a must.
I wouldn't rule out that a black dragon trapped under a ton of rubble couldn't eventually use his breath weapon to try to dissolve his way out, but I doubt it would be very effective. It surely ain't gonna be: "I'll huff and I'll puff and free myself in one round." Consider energy attacks do half damage, then apply the hardness of the material and every square has literally a ton of stuff on top of it. This might work if we are talking about a red dragon buried in an avalanche.
This is a GM call, but I don't think the monster being able to simply use a breath weapon in lieu of a STR check is really justified.
There's also the problem that using a breath weapon to dig out might simply make things worse, like trying to clear a cave-in with dynamite. This is a strictly flavor area; there is no RAW support for it.