As far as the burgers are concerned, what is cooking out of it depends on the fat content of the burgers - a whole lot more fat will cook out of a 75% lean burger than a 92% lean burger.

A skinless chicken breast is nearly fat free, so there is very little fat available to be cooked out of it.

All fresh meat also has moisture (water) content. I don't know whether they prepare chicken breasts for sale this way in the UK, but a lot of chicken breasts sold in the US are water chilled, which results in the meat retaining water from the chilling process, in some cases as much as 15% of the weight comes from the chilling process, during which the chicken is soaked in a saltwater solution in order to quickly chill it to a safe storage temperature.

At any rate, since all meat also has a naturally occurring moisture content, the cooking process will remove some of that too. How much of the moisture ends up removed in the cooking process depends on how thoroughly you cook it. If you cook it until it just barely reaches a safe internal temperature, then you won't lose nearly as much moisture as you would if you continue cooking it until the meat is overcooked and dry.

Since neither hamburger nor chicken breasts should have any carb content at all, the amount of liquid lost will not affect carb count. Higher fat hamburger will lose fat and calories during the cooking process, but the thing about hamburger is that if you opt for higher fat burger meat to save money, and you're using a cooking method which allows all removed fat to drain away, the total fat/cal content of the cooked meat isn't all that much different between 75%, 80%, and 85%.

If you're pan frying, and intentionally leave the cooked burgers in the pan to soak up some of the fat that cooked out of them, you can retain more of the fat and calorie content, and your burgers will be more moist from the retained fat.