This is the example provided in the campfirebackport.readme.txt:

mods.campfirebackport.addCampfireRecipe("both", minecraft:glass_bottle, minecraft:potion, 63);

When I reload the script I receive this error:

ERROR: Error executing Campfire.zs: Internal compiler error: int constant type is not an int

Removing the cookingTime allows the script to work. Is the script missing something or is the mod not looking for the cookingTime properly?

Looks like it's an issue with older versions of MineTweaker.
I made it work with MineTweaker 3.1.0 for 1.7.10, and I was able to replicate your problem with MineTweaker 3.0.10B for 1.7.10.
I thought more recent 1.7.10 modpacks were using the download from CraftTweaker instead of the download from MineTweaker, so I went with that one. I'll put something somewhere to let people know.

Well my face is red. I was unaware of the update to CraftTweaker. I went ahead and updated. Thank you for pointing that out.

I'm now getting a new issue. The campfire lets me cook whatever I want but after it produces one output any other output of the same item can no longer be produced.

Example: I cook a pork chop, it finishes and pops off. I try to cook any others and they just vanish after they finish cooking. Am I doing something wrong?

Edit: I'm using the default config for pork chops, no scripts, but the same issue occurs for every item I've scripted, too.

Edit 2: I'm a moron. I was using a later version of CampfireBackport. This new issue has already been fixed. Sorry for wasting any time. Please ignore.