Hi and welcome to the forum! I am glad that I can help you.
You would like to know the cause of the egg-smelling burps that you experience.
An egg smell appears due to a gas that contains sulfur. This may be common if eggs are consumed frequently. Burping appears when there is an air pocket trapped in the stomach (gaster). This air pocket may originate from the air that individuals swallow during speaking, eating or drinking. It may also appear due to consuming carbonated drinks. As the air pocket grows, it causes pressure on the stomach (gastric) walls, which contract and expel the air in a bubbling or loud sound(burping).
Given the data that you described, you may or may not be experiencing any medical condition. Avoid consuming carbonated drinks to reduce the pressure in the stomach. Reduce your weight to reduce the abdominal pressure that obviously makes your bellybutton pop out. By reducing the abdominal pressure, the pressure upon the stomach (gaster) will be decreased as well and burping will be avoided.
Please keep in mind that I provide medical information only. I am not able to diagnose medical conditions online. Please contact your doctor for further advice and information about diagnosis and treatment options.