why isn't there a place for a black man in the army?

He doesn't say in-detail what happened. But from the context, as well as the overall message of the film, he is likely implying that there is prevalent racism towards blacks in the army.

Well keep in mind this movie was set in the late 80s-early 90s. He believed they were sacrificing themselves for a government/nation that really didn't care about them. And a war like Vietnam the veterans weren't treated well and came home all messed up.

As someone from LA who's done nearly 20 in the Navy, I can honestly say the military is a far better option than just trying to get by in the same neighborhood as the clowns you grew up with. As long you aggressively use them and take advantage of the benefits many ignore, you'll get plenty out of the military life.

Because it doesn't make sense to fight for a country that doesn't fight for you.

It's a part that is up for debate because military can lead to great opportunity and benefit but Furious did not have a good run in the army so naturally he's against it.

So he may have felt that way during war times and for a conflict that was again up for debate.