Martin Luther King Was Not a Republican, He DESPISED Them

The modern right tries in vain to claim they are the party of Lincoln.
The GOP was the party of Lincoln in the mid-1th century, when it was a party of liberals.
The GOP is not, and has not had anything in common with Lincoln, other than a misleading party name, for over a century.
What MLK thought in the 60's of conservatives, both the regressive Dixiecrats, and the conservative GOP, especially the division of the party represented by the extreme right, and the fringies like the Birchers, remains true today.  The right is racist.
From the MLK biography, by  way of Think Progress:

The Republican Party geared its appeal and program to racism, reaction, and extremism. All people of goodwill viewed with alarm and concern the frenzied wedding at the Cow Palace of the KKK with the radical right. The “best man” at this ceremony was a senator whose voting record, philosophy, and program were anathema to all the hard-won achievements of the past decade.
Senator Goldwater had neither the concern nor the comprehension necessary to grapple with this problem of poverty in the fashion that the historical moment dictated. On the urgent issue of civil rights, Senator Goldwater represented a philosophy that was morally indefensible and socially suicidal. While not himself a racist, Mr. Goldwater articulated a philosophy which gave aid and comfort to the racist. His candidacy and philosophy would serve as an umbrella under which extremists of all stripes would stand. In the light of these facts and because of my love for America, I had no alternative but to urge every Negro and white person of goodwill to vote against Mr. Goldwater and to withdraw support from any Republican candidate that did not publicly disassociate himself from Senator Goldwater and his philosophy.

The Think Progress article went on to point out that King was pro-LBJ, noting:

David Garrow, who wrote a Pulitzer Prize winning biography of King, stated “It’s simply incorrect to call Dr. King a Republican.”
In 2008, King’s son Martin Luther King III said “It is disingenuous to imply that my father was a Republican. He never endorsed any presidential candidate, and there is certainly no evidence that he ever even voted for a Republican.” Garrow claimed there is little doubt King voted for Kennedy in 1960 and Johnson in 1964.

There was no justification for people of color to support conservative policies, not then, not now, not ever.  The efforts to deprive minorities and students and other groups that tend to vote Democratic of the vote will guarantee that the deservedly poor opinion held of conservatives continues for another extended period of time, possibly another 100 years.
King had a dream; conservatives have a nightmare vision of freedom, of the nation.