I think it would also be worthwhile for you to help coach your DH into coaching Jeff, if that's something he'd be open to.
Based on all of your descriptions, it can definitely believe that Jeff is acting out of inarticulate and ill-defined emotion, which is far more complex than "I don't want to feel like I have to give gifts." It sounds like it could very well be a situation of "Now that I no longer live close enough to see people often and don't have time to correspond with them frequently, I feel like the only way I can keep up my connections with people is through spending money and time on gifts, and that's too much for me to feel good about it."
It's hard to move far away from your family and your childhood home for the first time, and I think it can be particularly hard in your 20s, because you're still finding your "adult" feet, so to speak. I did it right out of college, and it took some pretty significant adjustments on my part. I've gone through a lot of angst about how to stay connected with family that was reasonable, sustainable, and still felt good and satisfying for me. To be honest, I'm not sure I've figured that out even now (7 years in), but I am at least more capable of rolling with it.
I think it would be fine to ask Jeff what the magic words are to tell him about family events and milestones without pressuring him on gift-giving, although I doubt he's ever going to be able to come up with them. Still, asking him (as many times as necessary) how to keep him in the family loop without inadvertently causing pressure to give gifts will make him think about it. And in thinking about it, he may realize what is at the root of his anger on this issue and be better able to articulate it.
Because truly, I don't think this is about the gifts. I think this is about the transition, and the gifts are a symptom. His end result may still be not giving gifts particularly often, but his figuring out why the expectation bothers him so much will help him be comfortable with whatever level of gift giving he arrives at.
So I'd say that your DH (or you, if you get the opportunity) should just not let the conversation be about the gifts any more than you have to let it be. Make it about what he wants you to tell him about, or what specifically bothers him about specific gift-giving scenarios. Just keep asking him to be more specific in general. And if you start going in circles or determine at any point that you can't help him any further, tell him that you aren't able to help him on this topic and so don't want to discuss it anymore (and then don't discuss it).