For the most part, negative calories are a myth. The claim from that website, that those foods have "a tremendous natural fat-burning advantage." is really hogwash. It is true that there are some foods, such as celery, that contain so few calories and high fiber that between chewing and digestion, you may burn an extra calorie or two more than they contain. But seriously, the amount is so SMALL that you would have to binge on celery nonstop all day long, and it would still take a long time to see any result from that. You have to burn 3500 calories more than you consume to lose one pound of fat. At least that's all that the so-called negative calorie effect can do. Many of the foods on those lists don't even qualify in the same way.
I've read articles by several dietitians that have looked at the list you posted. Most of the foods listed do not qualify because they contain a lot more calories than they use to digest. However, everything on that list is good for you, and is still low in calories, so it can only help to include them in your diet. But technically, some of the foods on that list may qualify because they do actually burn more calories than they contain, though it's usually just a couple of calories.
Here's the REAL negative calorie list, according to Marcia Byrd, clinical health educator at Kaiser Permanente Medical Center in Gilroyand, and it isn't nearly as appetizing,

Asparagus
Broccoli
Cauliflower
Celery
Cucumbers
Garlic
Green beans
Green cabbage
Iceberg lettuce
Onions
Radishes
Spinach
Turnips
Certains kinds of zucchini
Even with the updated list, there isn't really a fat buring potential. If anything, you can eat a lot of them and not worry about gaining weight.