I'm basing here ideology on position of Jared Taylor as expressed on website of American Renaissance. In Wikipedia he is referred as "white nationalist, white supremacist, and racist", while he claims that his position is merely "white advocate". As he was pointed out in other answers, I'd consider him as important enough. Moreover ADL calls him as "academic racist" who "has also spoken a number of times at colleges in the U.S., where students who have embraced a white supremacist ideology see him as a spokesperson for their cause", so seems suitable. In consequence this list of arguments should be rather internally consistent, but at expense of neglecting other views within such movements.
1) All ethnic groups have fully legitimate interests and is generally acceptable for other ethnic groups to openly pursue them
Racial loyalty or racial consciousness is normal and healthy. All non-white groups instinctively pursue their own interests, and legitimately so. It is only whites who have been taught that it is immoral to take even the most basic steps to ensure their survival.
2) Differences in outcome among ethnic groups are mainly result of genetic differences, which explains why prior attempts to ameliorate them failed
Regression toward the mean explains something that has always baffled the “sociologists:” children of low-income whites (and Asians) get better SAT scores than the children of high-income blacks. If environment controls IQ, the children of wealthy blacks should be enjoying the benefits of good environment. They are, but those benefits are meager and do not make up for the effects of heredity and the lower mean toward which black children regress.
There is also much ignorant shrieking about the “bias” of IQ tests designed by white men, but it is an odd bias that permits Asians to outscore whites. As Prof. Levin explains, a real example of bias would be a test of hand-eye coordination that involved only the right hand. Lefties could prove the bias of such a test by demonstrating their ability with their left hands. “If the races are equally intelligent,” he writes, “it should be possible to find a task intuitively requiring intelligence that blacks perform as well as whites.” No such task has ever been found.
Yes, I consider as interesting phenomena that person referred as "white supremacist" technically speaking makes an argument that in IQ tests Asians outperform Westeners.
This blindness leads to futile attempts to eliminate “achievement gaps” between the races and leads whites to accept the view that if blacks and Hispanics are less successful than whites, it is because of white “racism.”
3) Ethnic groups, including whites, have right to preserve and be proud of their heritage
Black rejection of not only the Confederate but the American heritage is clear in the removal of the name of George Washington from a public school in New Orleans. On Oct. 27, 1997 the Orleans Parish School Board, with a 5-2 black majority, voted to change the name of George Washington Elementary to Dr. Charles Richard Drew Elementary (Drew was a black surgeon who made advances in preserving blood plasma); the school itself is 91 percent black. “Why should African-Americans want their kids to pay respect or pay homage to someone who enslaved their ancestors?” asked New Orleans “civil rights” leader Carl Galmon. “To African-Americans, George Washington has about as much meaning as David Duke.”
4) There are differences in crime rates among ethnic groups and is rational stay away from crime hot spot
There are dramatic race differences in crime rates. Asians have the lowest rates, followed by whites, and then Hispanics. Blacks have notably high crime rates. This pattern holds true for virtually all crime categories and for virtually all age groups.
In 2013, a black was six times more likely than a nonblack to commit murder, and 12 times more likely to murder someone of another race than to be murdered by someone of another race.
5) Some US school admission regulations like "holistic score" seems to be intended to discriminate against whites and they oppose it
Two things saved UT from “resegregation.” The university immediately invented a new, “holistic” way to evaluate applicants, which piously ignored race but gave extra points for growing up without a father, being poor, not speaking English at home, etc. It was a transparent attempt to smuggle race into admissions decisions, but it didn’t work very well; an annoying number of whites were poor and fatherless and therefore got the leg up UT wanted to reserve for non-whites.
6) Already people who officially do not share views of white nationalists, take part in white flight in order to live in good neighbourhood (which somehow turns out to be mostly white), thus inadvertently support their point for ethnic separation
In the 1960s and 1970s, whites were generally fleeing blacks, but the great black migrations have largely come to an end, and whites have reestablished distance between the two races. In recent decades, it is massive, non-white immigration that most often drives whites from their neighborhoods, and continuing immigration only hardens the alien character of these places. No one believes that the arrival of yet more Haitians, Guatemalans, Mexicans, Jamaicans, or Vietnamese will somehow restore the former character of South Central Los Angeles or Miami and induce whites to move back.
The process works the other way. As their numbers increase, non-whites continue to expand into adjacent areas. Whites, many of whom fled their homes in the face of the first incursion, move away once again.
This, then, will be one of the certain effects of demographic change: More and more parts of the United States will become, for whites, essentially uninhabitable. It will be physically possible for whites to live with the Mexicans of Brownsville, Texas or the blacks of Camden, New Jersey but such places will be almost as alien and as uninviting as Oaxaca or Mombasa. They will actually be more uninviting. The people of Oaxaca and Mombasa like and admire white Americans, whereas those of Brownsville and Camden have a strong and sometimes violent dislike for whites.
Dreaming about ethnostate seems acceptable (when it's Black Wakanada), being worried about being about demographically dominated by other ethnic group seems acceptable too (when applied to for example Tibet or Kashmir). Thus in their own eyes their position it seems as moderate, rational self-interest, that each ethnic group should be allowed to pursue its own aims and there is nothing wrong in voluntary self-segregation, even if it may lead to splitting of the US.
EDIT: I tried to find something of him that's considered as top inflammatory, hateful and would unquestionably justify all the infamy. Apparently this his video about race and intelligence got took down by Youtube as such, so one may watch "the worst of him" and make their own opinion.