I am not aware of any neuroscience papers that link breathing and self-confidence. One review that touched on breathing in yoga was Jerath et al. (2006). They discuss how deep breathing increases parasympathetic autonomic activity. This can change the physiological state of the body, leading to changes in affective state.
One area of the brain that may be relevant is a patch of cortex called the insula. The posterior insula receives afferent information from the body and creates the initial sensory representation of body state (Craig, 2000). From there, the mid insula begins to integrate body state with other sensory modalities, such as visual and auditory information. This progression continues, until you have a highly integrated, abstract representation of interoceptive and emotional state in the anterior insula (Craig, 2009).
Activity in the anterior insula has been related to a wide array of higher cognitive functions, including self-recognition, emotional awareness, risk prediction, conscious awareness of sensory stimuli, temporal perception, attention, perceptual decision making, performance monitoring, and more. We know that relaxation techniques can influence activity in this area of the insula (Critchley et al., 2002).
It is well supported that controlled breathing changes the state of your body and activity in the brain, including the insula. These changes can also directly influence our cognitive state and cognitive ability. Changes in self-confidence may be a part of that, depending on how you might define self-confidence.
Craig AD, Chen K, Bandy D, Reiman EM. (2000) Thermosensory activation of insular cortex. Nature Neuroscience. 3, 184-190. PMID: 10649575.
Craig AD. (2009). How do you feel-now? The anterior insula and human awareness. Nature Reviews Neuroscience. 10(1), 59-70.
Critchley HD, Melmed RN, Featherstone E, Mathias CJ, and Dolan RJ. (2002). Volitional control of autonomic arousal: a functional magnetic resonance study. NeuroImage. 16(4), 909-919. PMID: 12202079.
Jerath R, Edry JW, Barnes VA, and Jerath V. (2006). Physiology of long pranayamic breathing: Neural respiratory elements may provide a mechanism that explains how slow deep breathing shifts the autonomic nervous system. Medical Hypotheses. 67, 566-571. PMID: 16624497.