What's the point of the quote anyway? Even if it's genuine it would simply show that Johnson, who was born in Texas, in 1908 had a crude sense of humor.

LBJ *was* a crass man, and his biographers don't shy away from saying so. (He once bragged that he "had Ho Chi Minh's pecker in [his] pocket.") He frequently used the N-word in private (again according to many sources), apparently just because he was used to doing so and didn't care who he shocked. None of that changes the fact that he actively supported and did sign the historic Civil Rights Act of 1964.

There appears to be no good evidence that he ever uttered the words in question, either just after the signing (in front of dozens of witnesses of both races, including members of the news media), or at any other time.

An Internet search reveals only a tiny handful of instances, all very recent, all far-right libertarian related.

Several large newspaper databases reveal no report of the "quotation."

And a search of thousands of digitized books at Google Books and HathiTrust.org turns up only a single reference, in Thomas D. Kuiper's "I've Always Been a Yankees Fan: Hillary Clinton in Her Own Words" (2006), p.150.

Kuiper's book, made up entirely of alleged quotations, was published by the tiny, right-wing "World Ahead Publishing" of Los Angeles.

This link shows that some quotations are unreliable or simply false. Kuiper himself does not vouch for his own book's accuracy:

http://mediamatters.org/research/2006/04/24/clinton-attack-book-publisher-used-fabricated-q/135487   

Until somebody provides an earlier and more reputable source for "Johnson's" words, we can ignore the attribution as bogus.