you have no medical insurance thru your employer? Can you get it?
I'm assuming you live in Los Angeles... Googling found this place:
http://losangelesmission.org/about/lamcc.html
It doesn't say anything about income requirements, however, if you are getting a service from them, and you are employed (and you are concerned about making too much money to be treated), if they do treat you, a donation would be in order. but not mandatory.
Confidentiality in clinics, I know refers to AIDS and HIV screenings and treatment, I'd imagine it would apply to all medical care as well.
The only time I have been to a clinic, was in NYC when I had to go for a TB screening before volunteering, and for an AIDS test before volunteering with babies. The AIDS test I as never asked my name, I was given a number. The TB test, I was asked to fill out a form, but never asked for identification so they dont know if the info was accurate