The way I would interpret black-and-white (when used as a verb) would be:
Don't put everything into [two] absolutes when you're talking with me.
Two common idioms are:
It's not black-and-white.
When something is either morally right or morally wrong, it's often said to be black-and-white, meaning that there are no shades of gray. For example, one might argue that stealing is wrong; it's simply not right to take what doesn't belong to you.
Someone else may counter, though, by pointing to the legend of Robin Hood. Though a thief, he was lionized, because he didn't steal for his own gain. Rather, he stole from the rich, and gave to the poor, who were being oppressed by the rich.
Some folks tend to see a lot of things in black-and-white, meaning they leave little wiggle room for "it depends on the circumstances." Others have more of a "shades of grey" worldview, meaning they understand there are exceptions to every rule, and certain situations call for going outside the lines.
black and white adj. (of a situation or debate) involving clearly defined opposing principles or issues
In a cry against racism, "Don't you black-and-white me" has a nice dual meaning. On one hand, it says, "Don't judge me simply by the color of my skin." But it also can mean, "Don't be looking at the world like there is never any room for compromise." Those two thoughts feed off each other pretty well.
That said, song lyrics and poetry aren't always easy to decipher; people have been inserting meaning that wasn't every intended into song lyrics for a long time. I believe one of the Beatles once quipped that whenever someone offered some interpretation of one of their songs, they might go along with it if they liked what the person said – even if those ideas were never in their minds when they wrote the lyrics. So there's really no way to know for sure, unless the author happens to be a member of ELL, and wants to chime in.
That said, the dictionary meaning of black and white can be ascertained, so I believe this answer may still be of some use for the English learner, even if I've read too much into the songwriter's work.