I'm having some trouble with the dragon-type enemies in the latter part of the game. The dragons' breath attacks tend to be "Game Over" for me. So that I can be prepared with Protect or Shell, are these breath attacks physical or magical?
The dragon breath attacks are physical based, with an element. What that means is that it can be both evaded (e.g. by ninja's Utsusemi or templar's Rampart) and stopped by spiritmaster skills. If you want to to use the white mage spells, go with Protect.
Just note that the dragons also use a skill that makes your party weak to its element, so you will take a LOT of damage, unless you somehow prevent it.