What would be the most important skills for career counselors and career coaches?

First of all as a career counselor or coach; you have to be positive and remain hopeful.

And you should be flexible enough to deal effectively with different careers.

Also, you have to learn every needed thing about the career you are counseling about, but the person you are dealing with; you have to know more about him.

I would say the most important thing is the knowledge about people, how to recognize skills and how to proof and valuate the knowledge.