It never ceases to amaze me how many people on TrekBBS criticize her and suggest she made up the whole thing. It doesn't seem that likely... and really, what does it matter? So long as the story doesn't change drastically, where she later says something like "After we met, MLK kept in touch with me, calling up after a watching an episode to chat, pushing me to get more screen time."
I don't personally criticize Nichelle. I find it slightly humorous how much the story has changed however, but I do find most thing's along those lines funny anyway. I personally believe that Nichelle is telling the truth and the story has probably changed over the years due to actor's being exposed to many great people and renowed people and perhaps having to focus their memory on stuff like scripts.
I do think Nichelle met MLK and I do think that he was impressed with the show and how it depicted hope for equality. Even if Trek was set in the future it still showed that racial equality would happen eventually. You can't count out the impact that the show had on civil rights at the time either. Many episodes dealt with racial issues and conflict, but they also delivered solutions and compromises. LTBYLB probably delivers the best example when Kirk and Spock are baffled by Bele's statement that Lokai's people is white on the right side and that is why they are enemies.