I'm experiencing constant watery green diarrhea, frequent burping that taste like rotten eggs, abdominal bloating and gas, all after eating a burger.

GERD. You may have gastroesophogeal reflux disease (gerd), with an upset stomach as well. Certainly avoid trigger foods high in fat-like the burger and try increasing fluids and taking an over the counter acid blocking medications such as Prilosec with maalox. Eventually, this is a warning sign that you need a good GI evaluation to get properly diagnosed. Best wishes.
Flu Season. During flu season the stomach flu is very common. Also, infections such as from food can cause these symptoms. Given that your symptoms came after eating a meal that could have had undercooked meat i would see how you feel tomorrow before contacting the doctor. That is, unless you do not get worse symptoms.