Dear Always Anonymous,
I've been going to a therapist on and off for years and I've never been formally diagnosed with anything. Primarily because, while I have experienced symptoms consistent with depression, anxiety, etc., I am not meeting the DSM-IV requirements for an actual diagnosis of Major Depression. Most people, including those that are prescribed anti-depression medications, won't actually meet the requirements that are listed in the DSM-IV. The point is that it's very rare that someone is actually diagnosed, and most therapists aren't trying to figure out what label to put on you, but rather they are trying to help you with whatever you need help with.
I personally love the CCC and I have had a couple of wonderful therapists there. I am especially grateful for them because most of them have an LDS background (or are very familiar with LDS beliefs), as this can help tremendously when your talking about your life, or beliefs, or struggles, or whatever it is you want to talk about. That being said, I get what Mico is saying, and the lack of complete anonymity can be a bit stressful at the CCC, so I will back up what she said about looking off campus.
Keep in mind that if you do go talk with a therapist/counselor, you are not publicly admitting anything. You are privately admitting something to an individual who's entire job is to keep your secrets and care for you. They are not there to judge, criticize, or "label" you in a negative light. They are not going to tell you you are crazy, or messed up; their goal will not be to try and fix you. Their one and only goal is to give you a safe place to talk about the things that are bothering, worrying, or stressing you out.
If you're still concerned, then test the waters. Go to therapy and don't talk about your major concerns on the first visit. Talk about school. Talk about roommates, friends, family. Talk about a small worry that you have. By all means, take time an build a relationship with your therapist first, before trusting them with your real secrets. There is nothing wrong with that.
As someone who carried her burdens for many years alone before getting help, it is amazing how much telling even just one person will help.