AP-
"To spare us needless time and effort, read the law yourself. There had been no earlier qualification regarding race. Creoles and other free persons of color belonged to the militia because they were citizens under the Treaty of Paris, Apr 30, 1803. The militia law of Jan 23, 1862, extended enrollment to include residents, not just citizens, but restricted it to white males. In response European men living in New Orleans and surrounding parishes organized the European Brigade of Louisiana militia."

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"Every free white man, between the ages of eighteen and forty-five, who has resided within the limits of the state for sixty days, is subject to militia duty."

Daily True Delta, November 22, 1861 (page three/"Home Department" article)

I have the full text of the militia law that was in effect in 1861. There is no provision for free blacks.
The NG volunteered under a law passed in December 1860.