I believe there is a link between calisthenics and meditation. Some static holds in calisthenics can induce a meditative state. I’d describe that state as one of being present in the moment, having a clear yet focused mind and strong awareness of the physical body.
Holding a handstand (and frog pose) has made me enter such a state. I was upside down balancing a handstand on my parallettes when I found myself looking intensely at a single point on the grass in the park. My hands were gripping the bars tightly and I was aware of my body, particularly my legs which I had to keep straight, core was engaged and pelvis tucked. For a second or two, being upside down felt effortless. I could sense the atmosphere in the park and my breathing was steady. I felt at peace and in harmony with my surroundings. Succeeding at balancing a handstand created a feeling of balance within me.
Therefore, I believe calisthenics is not only good for building muscle and strength but also for coping with stress, improving concentration and becoming more present. I must admit, however, yoga would require significantly less effort to help you reach a state of harmony. Doing a headstand or balancing on one leg are nowhere near the challenge of doing a handstand but I guarantee yoga will perform miracles for your mind and body.