Every time my wife and I visit Anguilla, we always stop at Palm Grove (aka Nat's Place) for lunch.
Palm Grove is located on Junks Hole, on the east side of the island. After you leave the main road, the road down to Junks Hole is unpaved and very rocky, though it's better than it was when I first started going there years ago. When you finally arrive, you'll see a run-down looking shack, but don't let that fool you -- the food is great!
We usually visit Anguilla at the beginning of the slow season (May or June), so there are typically few (if any) other customers when we have lunch at Palm Grove. On this visit (in early June), we were the only ones there. Nat, the owner, wasn't there, because he wasn't feeling well that day -- a man named Adrian was there in his place. We had the grilled fish, as we usually do, and it was great as always. The meals come with johnny cakes, which are always great too. We ordered rum punches, and Adrian told he didn't have any fruit juice on hand, but he immediately ran out to a store to pick some up for us. (They were good too.)
Like the restaurant, the beach at Junks Hole is always pretty empty, so you have the place to yourselves. Other than Palm Grove, there are no business or homes near the beach, and it's not maintained as well as some of the others on the island, so there can be a bit of debris washed up on shore. However, the beach was relatively clean on this visit. The section of the beach near Palm Grove is rocky, and the snorkeling is supposed to be good in that area. I tried snorkeling there on a previous visit, but it wasn't that good (at least on that day). so I haven't tried again since. If you walk down the beach away from Palm Grove, it gets much less rocky and better for swimming.
You really ought to give Palm Grove a try if you haven't been there yet. It's a unique experience that you won't have anywhere else, and the great food will make it worth your while. I've posted a few photos to give you a feel for the place, but you really need to experience it in person.
This review is the subjective opinion of a TripAdvisor member and not of TripAdvisor LLC.