First, it's entirely historically correct to call them "Negro spirituals" but it's become the accepted practice to call them "African-American spirituals" or simply "Spirituals".
I agree with other commentors who say that it's about the way that it is performed. There is only so much you can do to notate a spiritual on a piece of paper. An effective performance would include elements like a very heavy emphasis of beats 2 and 4 in 4/4 time even if the piece is a capella vocals--especially if the piece is a capella vocals.
In spirit (pun intended) let me offer my opinion that, especially if you are not African-American yourself, it is important to approach the source material and the subject matter with great reverence and humility. If nothing else, just respect the fact that these anonymous folk songs are a tremendously important part of American music history, so don't patronize or belittle them.
Let me share an anecdote from my many years of singing choral music. A college choir I sang in was preparing for a concert with a guest conductor. The guest conductor was not going to work with us until the actual concert. We needed to learn one well-known published arrangement of a spiritual. So we rehearsed with a student conductor. Everybody was white, or at least not African-American. My choir and our student conductor rehearsed the piece and learned it with a very heavy emphasis on the 2 and 4, at a slow tempo, with a deep groove.
At the concert, the white director conducted it at a fast tempo with a perfectly even classical beat pattern with the heavy beat being 1, and the secondary beat being 3. Exactly the opposite of African-American music. It sounded so laughably bad that when the concert ended, we were all laughing about it, and bitterly.