Megastability and coexisting attractors between periodic and self-excited chaotic attractors in the Josephson junction snap oscillator (JJSO) are reported in this paper. Two different cases are considered for this study. In one case, the JJSO is driven by a direct current (DC) whereas in the other case, it is driven by an alternative current (AC). The JJSO is implemented using a microcontroller to verify the numerical simulation results. The numerical simulations and microcontroller results have a qualitatively good agreement. Finally, an encryption scheme based on the modified Julia set, deoxyribonucleic acid (DNA) principle, and the sequences of the chaotic JJSO in this work are used to build a secure key to secure medical images. The plain colour image is decomposed into a blue component. The generated secure key is applied to one component to obtain the corresponding sparse components. Confusion keys are obtained from the chaotic JJSO to scramble each sparse component. Besides, the chaotic sequence is used to compress the confused sparse matrices corresponding to the blue components. Lastly, the component is quantified and a diffusion step is then applied to improve the randomness and consequently the information entropy.