Within a 3-day period at a large hospital 5 patients undergoing hemodialysis developed fever and chills. three gram negative bacilli, pseudomonas aeruginosa, escherchia coli, and proteus vulgaris were isolated from 3 of the patients and from the dialysis system. why might all three bacteria cause the same symptoms?

All three bacterial species are gram negative. Gram negative bacteria tend to survive better in Internal cavities of the body as they do not have thick cell walls like gram positive bacteria, so tend to "dry out" in dryer conditions like on the skin. 
So I would say the urinary tract provides them with a "good" living environment to thrieve and cause infection. They are all opportunistic pathogens. They are also all found naturally in nature in soil and plant roots so maybe this has something to do with it? 

You did not show us the image of the burger, but i shall try to answer as best i can.  when purchasing food, always consider the ingredients as well as the person who prepared it. consider if they were hygienic people or not, in case the burger was infected with bacteria. no mortal should place an infected burger into their mouth.