I'd say, no, we don't need a black Hawkeye. Especially not to rival John Stewart.
If you'd really go that way, the best counterpart to John Stewart is James Rhodes.
I wouldn't like a black hawkeye for a couple of reason.
First, we already have two active Hawkeyes and there's still so much potential that'd be a shame.
You also have to consider Hawkeye is an outdated character. His concept (a bowman) did work in the 40's to the 60's but there is not much credibility to this character now. Same for the Green Arrow family. To "update" it you have to go the MCU way or the Ultimate way and make him a killer. A mercenary. A soldier.
I'm against super heroes killing. And I'm a bit fed up with black military or mercenary heroes. ^^