But critical thinking is not just about mental skill. It also requires the adoption of certain moral values without which those skills we talked about before might be put to bad or unfair use. Some of these values include integrity, honesty, humility, and the ability to recognize when you've made some terrible logical blunder. Of course, when you simply re-affirm bad arguments, even though they have been shown to be fallacious, defending them is only going to require that you produce even worse arguments, rife with fallacy and dishonesty, so even if you don't believe in the values above, it would still be better to cut your losses and quit while you're ahead.
Anyway, in my critical thinking classes I usually assign a project that requires students to identify instances of certain informal logical fallacies as they occur in the real world, and the following is a two-minute sample based on the now infamous episode in which Bill O'Reilly thought he could "prove" the existence of God based on his own ignorance (just go to the second slide to see the logical analysis).
If he had just said "my bad, I made a mistake, that really was a bad argument" he probably would not have lost the respect of so many people...