I originally wrote a comment but felt myself wanting to add more coming from the perspective of someone who has had the same thoughts as your coworker and lives with mental illnesses.
I think he may be depressed
I wouldn't begin to assume your intern is actually depressed until he starts to exhibit more related signs. The situation doesn't seem terribly serious at the moment, it could be family trouble, a relative passing away, or just something that has them upset (but sometimes it is hard to know). When you start to notice things like; not showing up for work (repeatedly), exhaustion (likely from lack of sleep), lack of communication, isolation, lower work quality than normal; that's when you should start to be more concerned. It's a tricky spot to be and deciding to encourage them to get help vs helping yourself is hard. At the end of the day, it's not your job to be a medical advisor if they need one, but that doesn't mean you can't be supportive.
Your best bet is to just listen and respond when appropriate, for right now. If the situation seems to become more serious, that's when you can begin suggesting they seek help outside of work. (They may already be seeking that help).
If you start to feel overwhelmed by the situation or if they continue this behavior with no change, then it starts to become a different discussion on whether or not they're just saying them because of social anxiety/awkwardness, etc.
Note - be careful with how you approach it as others have said. https://www.mentalhealth.gov/talk/friends-family-members Check out this source for some tips (even as a coworker some of these apply).
"I am not sure if I should be worried about him or just let it go"
It sounds like this person trusts you, so first and foremost, continue to listen if you can. You don't always have to respond or have an answer, they may just be shy or introverted too, and you allowing them to practice talking can be helpful. I would recommend (as others have) bringing up topics of your own interest to talk about during breaks to change the subject, you might find out they've got other interests and they're just not a great conversationalist (yet!).
Ignoring them when you may be the only person they feel they trust at work can cause the person to close up more or otherwise feel worse. However, as I said, It is not your job to be responsible for their mental health. You're in a position of support, not to provide treatment or actual medical advice.
"ask him to stop behaving like a teenager"
This behavior isn't necessarily exclusive to teenagers and younger adults. Mental illness, pessimism, etc can impact anyone of any age.
"I didn't expect that from him"
You've only known them a month, I think that's just a bad assumption.
Personal Note: I personally choose to not bring any of these things into work even if I think about them, but I'm someone who has sought help and learned how to cope and adapt with these things. They may not be there yet but that doesn't mean they can't get there.