Why is it significant that the town is named Greenwood and that the other briefcase comes from Shad Whitmore's shop?

The significance of these names is open to interpretation.
Both are mentioned in the Battle Royal scene in the novel's first chapter, which is sometimes extracted from Invisible Man and read as a short story. Greenwood is the town that the narrator comes from. The MC announces him as "the smartest boy we've got out there in Greenwood." The most obvious reference is to the bustling black community of Tulsa called Greenwood, nicknamed "Black Wall Street," which had been destroyed by white people. Ellison was a native of Oklahoma City, so he would have known about the community and its destruction in riots. The Battle Royal scene reflects the white supremacy that destroyed Tulsa—a power that could cause black people to try to destroy each other, as in the scene, or one that could wreak destruction, as was the case in Greenwood.
Shad Whitmore is the name of a local tanner, or someone who makes animal hides. The MC hands the narrator a briefcase that comes from Whitmore's shop. The briefcase is made of calfskin. The condition of the narrator could be interpreted as akin to that of the sacrificed calf. He, like the calf, is very young and is forced to turn his body into an object for someone else's gratification. He is reduced to an animal state and nearly killed.
The name could also be a play on the ironic phrase "shade white more." By participating in the event, the narrator commits an act to make himself more palatable to whites, further conforming to their expectations, as his grandfather had. The contrast between "shade" and "white" parallels the symbol of Optic White paint later in the novel—a shade of bright white paint mixed with a drop of black paint. This could be read as both a comment on racial admixture, which existed despite notions of white purity, or as an understanding that whiteness depends on blackness as an idea to counter itself against.