This might sound a bit high-flying, but I've come to the notion that posture is really a reflection of multiple interrelational layers of personal habits you are holding. These habits include subconscious/reflexive muscular or structural tension and relaxation (eg. you have tight traps and neck area or sleepy, inactive glutes), long periods of out of alignment isometrics (eg. you are sitting on a desk, tensing your traps and letting your butt go to sleep) and different psychological states (eg. you are constantly afraid, hyper-alert, fight of flight and stressed out, causing you to tense up, or you are lethargic and phlegmatic, causing you to overly relax). This is based on what I've learned during the past few years of research by reading up on the topics of mobility, flexibility, tension, posture and the like, and doing empirical research with a sample group of 1.
That said, I don't think calisthenics alone will "fix" your posture, but it can be helpful. Learning both how to generate tension (Strongfirst is the place for that) and how to relax will help. Moving your body in versatile ways, both powerfully and with grace, will help. Being aware of your body alignment and posture during your day will help. Understanding your psychological states of mind and working on those will also help. The bottom line is that no single modality will "fix" the posture, which is a collection of habitual layers, but instead one must address the entire collection.
On a side note, I once met a person who had undergone an Ayahuasca ceremony some time before. He told that before the ceremony he felt very uneasy and uncomfortable in his own body, and that the Ayahuasca showed him how to be in his own body. This changed his posture, gait and the entire way he moved. Not saying everyone should do it, but definitely goes to show the strong connection between one's posture and state of mind.