I doubt you will get chubby switching to calisthenics. The only way you’d gain fat though is if you just don’t go as intensely and end up burning fewer calories but keep eating the same. As long as you keep the intensity up you will probably get leaner. But there’s no reason you can’t lift and do calisthenics. There’s actually a thread about that that’s been active in the last day, you should check that out.
I have a friend who is an avid crossfitter but since training with me has added more advanced calisthenics to his training and, after training with me, sees real tangible benefits to his lifts. He sees it as a big advantage cuz most crossfitters don’t concern themselves with training for anything beyond muscle ups with huge kips.
I’d also suggest sprinting or some other form of intense cardio if fat loss is one of your primary goals.