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.