When you are acting as the representative of a nation or a group that shares your beliefs and for which you would die, you can use this loyalty to strengthen your own resolve. Egwene was basically stating what the Amyrlin Seat and the White Tower stood for, and how she, Egwene al'Vere would not consider any possibility of failing.

I liked that little section, by the way - it had more to do with Mesaana being a second rate Forsaken than anything. I can't really see Ishamael/Mori or even Lanfear/Cynd being that easily defeated as they would just laugh and talk about how they undermined the White Tower before or freely use it as their vessel for evil.