I have gone threw many muster roll records of entire Trans Mississippi regiments. As a results I have noticed that units raised west of the Mississippi were not particularly segregated. That Blacks, White, Hispanics, and even Native Americans soldiered side by side with even Mexican and in one case Chinese nationals.
The same was seemingly true of religious teachings, Catholic, Jews, Protestant, Baptist, Methodist, even Buddhist.
The explaination that I have heard put forward for the lack of forming separate companies of blacks, from a political/military standpoint, was that the southern leadership did not want the war to become a war over the Blacks. To the southerners it was a war for Independence, not for keeping or freeing the slaves.
Although there were plenty on both side who tried to make it a war over the slaves. The south went to some lenght to keep the war a "white mans" War.
While Cleburne's purposal was very practical and even endorsed by several including Robert E. Lee, the two main objections were, Arming slaves, and keeping the moral High ground, in the southerners eyes, of fighting for Independence and keeping the war a war between "white men".
As we see history did not go that way and the Northern government made the Blacks a focal point of the war.