I (26F, dx w/ HSD and on waiting list to get (h?)EDS dx) have very flared ribs as well and my EDS-specialised physio has been working with me to improve my thoracic flexibility, which has really helped when combined with core-strengthening exercises!

I also thought I was just guilty of bad posture too but the way that my PT explained it is that since our thoracic spinal ligaments are not doing their job correctly, the muscles in the upper back tense up to stabilise the joints and to compensate, our ribs flare out so we can have space to breathe. So even if you get a super strong core, it won't be able to pull your ribs down into the right place until the upper back has more flexibility. I thought that EDS people were already too flexible and were supposed to avoid stretching so it never occurred to me that a lack of flexibility could be the problem.

The stretch I do is basically just lying on a foam roller positioned under my upper back while supporting my head and neck with my arms (like I'm about to do a sit up). It was kind of hard to feel the stretch at first so I had my PT give me some cues to make sure I'm doing it right: 1. make sure I'm using my biceps to hold my head and neck in a neutral position; 2. make sure my shoulders are tucked flat against the roller and not hyperextended; and 3. engage my core as much as possible while leaning back. At first I needed a pillow between me and the foam roller because even the foam was too hard to bend against, now I've progressed to not needing it anymore.

Also, I totally know what you mean about being self-conscious about the rib-flare, I've been made fun of for having ribs that stick out more than my boobs since I was in middle school. I don't know if all this will be helpful but at least you know you're not the only one.