If she is truly as you describe, I wouldn't want to deal with her on any level.
We all have days where we want to cry because we are simply so stressed out that you have to get the emotion out somehow. At least I know I get like that on occasion. My solution is to hold it in as long as I can (hopefully until I get home) and if I can't, I go to the restroom and let it out. I also work in a company that has only four females out of 100 employees so I am always conscious of being seen as overly emotional. My co-worker on the other hand has been known to cry at her desk, but only when she is going through a personal crisis or is just simply to tired to cope. And neither of us has ever broken down in front of any of management when being criticized. (We wait until they walk away.)
To answer your question, yes I think this behavior is destroying her credibility. If she cannot handle day to day stress without breaking down into tears, how could I ever believe she could handle the high pressure items that management deals with on a daily basis.
Michelle
