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Of course. It matters either way too. The pendulum does swing both ways.
http://www.the-fanboy-perspective.com/

Based on the cartoon and modern iterations of the character, his "blackness" has little relevancy. It's mostly his pre-cartoon days in the comics, especially his roots in the 1970s. The short lived
Green Lantern: Mosaic
series in the early '90s is probably the best character study of John Stewart before the cartoon semper fi-ed the character up.

I mostly know John from the cartoon. And going off of that, I never thought his race was all that important.
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It's mostly in the character's beginnings. Stewart was in some ways the stereotypical "angry young black man" with authority issues and a chip on his shoulder. Hal kind of has that assessment of him early on and Stewart considered Hal's outlook bigoted. Hal at one point is scolded by an old black man because Hal has done so much work for the "purple skins, orange skins, and blue skins" (aliens) but nothing for the "black skins" on his own planet. Green Lantern comics got real preachy and focused on social issues in the '70s when they paired Hal Jordan with liberal loudmouth Oliver Queen (Green Arrow). It was out of this that the character of John Stewart emerged.
Of course a lot about the character of John Stewart has been retconned and rewritten. The cartoon introduced the marine concept which the comics eventually ran with. It's kind of hard to have a soldier guy have authority issues.
Originally Stewart was a soulful guy and an architect who was about building things, including relationships with people. Then they crammed "Marine" into all of that because the cartoon version became so well known. I think the cartoon just wanted to make Stewart something more exciting than an architect since they were using him to totally replace Air Force test pilot Hal Jordan.

Outside of John being DC's first black superhero, how does his blackness inform his character? I'm genuinely curious, as I didn't know that it did.
For the record, my opinion on this is that I don't really care. All things being equal, I'd prefer them to find an actor that somewhat looks like what the character is supposed to look like. But race changes don't normally bother me.
I'm just aware of the fact that a white character's race is far more likely to be open to change than a black character's race. And that could be because a lot of black character's "blackness" informs their character. But in John's case, I didn't really think it did. He's not like Shaft or Luke Cage (as far as I know).
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I don't really see the point in doing that, but I would take Hal Jordan in any color over John Stewart in any color, all else being the same about the characters. I don't mind John Stewart's inclusion in a Green Lantern movie franchise, but I'm not at all interested in one that omits Hal Jordan.
Stewart's race does inform the character somewhat, so making him white would alter his perspective a bit.