Both are great companies, and offer very similar compensation, bases, equip, etc.
I can only speak for Eagle, but it has been steady and stable for the past decade. No telling what tomorrow holds, but we will find out then. The training department is top notch. The training is intense and they expect you to study hard, but you will come out as a better pilot than you went in. Eagle is hiring 600 for 2011, classes every two weeks and all bases/equip except LAX.