Which is more natural to say in Japanese? Do they have slightly different connotations?
- 食べないつもりです。 (I intend not to eat.)
- 食べるつもりはないです。 (I do not intend to eat.)
Also, are those translations fairly accurate?
They have different structures as you've translated, and both forms are used as often as the other unlike English. If I reword them to be clearer:
...and don't forget:
which is the negative of 食べるつもりです in the usual sense i.e. negation of the verb.
As you can see, 食べないつもり（だ） is used when you have a definite will not to do it. On the other hand, the interpretation of 食べるつもりはない is open to what you want to contrast/emphasize using は in each context:
Hey, there's a nice cake in the fridge.
It's for the party so don't eat it.
Oh, I didn't mean it. (= It's not that I intend to eat it.)
Don't you want to eat fugu?
I'm never going to eat it because I don't want to be poisoned.
食べるつもりはないです/つもりはありません is usually safer and more natural. 食べないつもりです is grammatically perfectly correct, but it can sound direct and harsh. People tend to use the latter if they clearly want to express a concern or hatred about the food.