Be careful. You don't really know this person. He's not a friend. You don't know if it's safe to share your experience with him. I definitely wouldn't share your PTSD with him unless you are comfortable with *everybody* in your workplace knowing about it. I'm normally comfortable with sharing a general sense of my struggles in life (I think it's important), but work situations are different.
It could be that your coworker shared this so easily because he has bad boundaries. You said he's annoying, and that's a clue that he does. Of course I don't know, but he could. People with bad boundaries can be challenging to deal with. Part of bad boundaries can be sharing inappropriate things with other people. For example, he may share your PTSD with someone else to try to bond with them. Sometimes people do this to create drama triangles. For example, he could tell your supervisor that you have PTSD and that it's affecting your work performance, because he wants your job or he's jealous of your success. If you tell him, something like this *could* happen. If you don't tell him, it won't happen.
If you really want to tell him for your own personal reasons, then I'd suggest getting to know him better. Ask him for a coffee. Eat lunch together. Stop by his workspace for a chat. Become friends, in other words. Then divulge your history. But it doesn't sound like you're interested in him personally like that. Don't do it for him, whatever you do. Do it for you.