The"rotten egg" odor comes from sulfur. Some of these volatile sulfur compounds are what give breath its bad smell.
The overwhelming majority of breath odors arise from the back of the mouth. (Only an extremely small percentage of bad breath problems are really due to stomach and digestive problems....)
In the mouth, bacteria feed on mucus and food particles that make those "stinky"sulfur compounds.
Be very vigilent about brushing and flossing and brush your tongue (as far back as you can before the gag reflex takes over!)
Since you seem to only notice this when you burp, maybe this is a stomach issue in your case. The odor comes from sulfur made by bacteria in the large intestine. Has anyone close to you commented on your breath in general?
You could try to reduce your intake of gassy foods, and also meat.
zuzu xx
PS-
Try this test: Take a Q-Tip and take a "swab" of the back of your tongue. Then smell the Q-tip... Does it have the rotten egg odor? If not, your breath in general is okay!