I agree, there is no need to disclose the details when they don't need to know. Dealing with depression to seek treatment is a legitimate need for time away from the workplace. If you are dealing with a chemical dependance along with it, they don't have to know that.
What matters is that you are a proactive employee who wants to get well. That fact right there is telling of what kind of a person you are aside from depression/chemical dependancy.
I'm so glad that your employer has an EAP. Please put it to use and get well! Look forward to happier days ahead!