I'm curious about a couple of things. Since your son doesn't get a report card, how is he informed of his progress or growth in a particular class? A kid in a regular public school sees that he got an A, or a D, and can compare from semester to semester, year to year. How does your son see if he's becoming proficient in a particular area of the curriculum?
And is it possible that your son is just trying to blend in with other students? Perhaps he excels in math and some of his friends are more average students. Often, teens don't want to be singled out as the class genius, due to potential teasing or bullying. Maybe you could encourage him to not call himself - or anyone else - stupid. Help him learn to be more objective, like saying "wow, this chapter is harder than I expected" or "graphing these equations is really confusing".
And when he calls himself stupid, what is your reaction? Sometimes we as parents tend to try to soothe, console, comfort. We ask "why do you think you're stupid?". But sometimes it's more helpful to simply state the facts. Your son says "I'm too stupid to get this", and you reply with "you know you're not stupid, but this math chapter is challenging. How are you going to master it? Do you need to memorize the procedures, or ask the teacher for help, or just devote more time to it?" If your son says he's dumb, simply state in a non-questioning way "you're not dumb, but memorizing all the German verb declensions is something everyone struggles with if they're not a native German speaker". Phrase the difficulty in terms of the assignment, not the person who's faced with mastering the assignment.
Tell him you're supportive of his efforts to learn and master the curriculum. Demonstrate your own efforts, like saying "filing our income taxes can be really frustrating, but I'm going to just go slow and steady and carefully" or "well, I tried making a caramel flan but it didn't set, so I'm going to try another recipe for it and do it again. I see where I made my mistake last time and I'm sure that with practice I can improve". Make sure that when your caramel flan collapses you don't say "wow, I'm such a horrible cook!".
From what I've seen of my daughter's major depressive disorder, her worst times are not when she expresses an evaluation of herself, but she shuts down silently. Your son saying that he's stupid seems to mean that he's able to assess himself, although not helpfully. The signs you mentioned, like loss of appetite, loss of interest, sleeplessness, often aren't accompanied by words, like saying "I'm not hungry". Instead they simply stay in their room throughout dinner. The ice cream that would usually be devoured overnight has been untouched for days. They don't usually say something like "I don't feel like playing soccer anymore", but you notice that the soccer ball has been sitting in the same place in the closet for a couple of weeks when previously it was your kid's constant companion and you always were complaining that the soccer ball got left in the doorway, in the entry, on your kitchen floor. Of course, that's not a professional opinion, just my own observations.
Its great that you're concerned and involved and interested. Sounds like you're doing a great job as a mom!