Those critiques have to exist first.

Alright, take the last chapter of the book, where this thesis is most evident. Sakai discusses the tactics and strategy of Black-White alliances, and he mentions the case of the 1890 union United Mine Workers (UMW), where black workers were the majority and white workers conspired against black workers to deprive them of their jobs and power in the unions:

Both Euro-Amerikan and Afrikan miners wanted tactical unity. However, since they had different strategic interests their tactical unity meant different things to each group. The Euro-Amerikan miners wanted tactical unity in order to advance their own narrow economic interests and take away Afrikan jobs.

Sakai shows how over decades, black worker were systematically removed from the UMW over time in several regions. His research is very enlightening indeed. However, his conclusion is as follows:

The entire example of attempted tactical unity shows how strongly the oppressor nation character of both the settler unions and the settler "Left" determines their actions. The settler "Left" tried to reach an opportunistic deal with reactionary labor leaders, hoping that Afrikan workers could be used to pay the price for their alliance.
(...)
So we see that tactical unity is not just some neutral, momentary alliances of convenience. Tactical unity flows out of strategy as well as immediate circumstances. Nor is tactical unity with Euro-American workers simply the non-antagonistic working together of "complementary" but different movements. Even the simplest rank-and-file reform coalition inside a settler union is linked to the strategic conflict of oppressor and oppressed nations.
(...)
What is important about these case histories is that they should push us to think, to question, to closely examine many of the neo-colonial remnants in our minds. "Working class unity" of oppressor and oppressed is both theoretically good, and is immediately practical we are told. It supposedly pays off in higher wages, stronger unions and more organization. But did it? (all emphasis mine)

And, finally:

The thesis we have advanced about the settleristic and non-proletarian nature of the U.S. oppressor nation is a historic truth, and thereby a key to leading the concrete struggles of today. Self-reliance and building mass institutions and movements of a specific national character, under the leadership of a communist party, are absolute necessities for the oppressed. Without these there can be no national liberation. This thesis is not "anti-white" or "racialist" or "narrow nationalism." Rather, it is the advocates of oppressor nation hegemony over all struggles of the masses that are promoting the narrowest of nationalisms - that of the U.S. settler nation. (my emphasis)

It's been a while since I've read Sakai, and I should note that I haven't read the work in full, but having read through most of it, I don't recall Sakai ever mentioning his definition of "nation", which he uses continuously throughout the whole book. I may have missed it in a single chapter, since I tend to read books non-linearly. To avoid any mistake on my part, I will make my definition of nation very clear, based on the outstanding Stalin work on it.

A nation is a stable community of people which shares:

1. A common language, but not necessarily a single language;
2. A common territory;
3. A common economic life, a cohesive economic bond;
4. A common culture;

Stalin attributes the term "nation" to the synthesis of these multiple determinations. Since Stalin, this remains as the most comprehensive Marxist theory of the nation and the national question. This is a far cry from the indiscriminate use of the term "nation" by Sakai. But you can deduct what Sakai treats as "nation" by the previous passages I mentioned here.

There is one thing that Sakai largely neglects and even ignore. Which is the question of ideology. Sakai treats white supremacist ideology and white labor as one and the same manifested as the "Euro-Amerikan nation". As an analogy, it's as if you looked at patriarchy and treated it as an aspect of the "Male-Patriarchal nation." If we consider the Marxist understanding of the nation, we can easily notice how the "Afrikan" and "Euro-Amerikan" nations have in common a language, territory, economic life and culture, perhaps with a few particularities on the culture one. But both constitute a single nation according to Marxist theory of the nation, especially because of the shared economic life.

One thing produced in Alabama is necessary to produce something else in Ohio, which is the essential part of a commodity assembled in California to be sold in the whole US (abstract example). The US is an integrated and indivisible whole, like every nation. Besides the national question, Sakai promotes the idea of "national liberation" of the "Afrikan nation." Taking that into consideration, what is the common territory and economic life of this "Afrikan nation" that is distinguished from the "Euro-Amerikan nation" so that the "Afrikan nation" can achieve its liberation? The more you question it, the less sense it makes.

Then in the last paragraph, Sakai advocates that considering all of that, white people cannot constitute a proletarian class. This is outright anti-Marxism, because be it a black person or a white-supremacist racist piece of shit, it doesn't change the relations of production. It doesn't change the fact that both are exploited, albeit certainly under different degrees, and the majority of blacks and whites do not own the means of production. It doesn't change the fact that there are also black exploiters, like Beyoncé with her Ivy Park fashion sweatshop in Sri Lanka paying $6 a day to produce clothing sold for more than $200. Or black agents of imperialism like Obama, Kamala Harris, and so on.

Now, to answer your comment:

Looks like someone forgot to read the book.

Did you read the book? Check out this passage, from chapter 4. It makes more evident how Sakai treats white supremacist ideology with white labor as one and the same:

What was the essence of the ideology of white labor? Petit-bourgeois annexationism. (...) The ideology of white labor held that as loyal citizens of the Empire even wage-slaves had a right to special privileges (such as "white man's wages"), beginning with the right to monopolize the labor market.
(...)
Since the ideology of white labor was annexationist and predatory, it was of necessity also rabidly pro-Empire and, despite angry outbursts, fundamentally servile towards the bourgeoisie.

How can someone read those excerpts and not see from a mile away the intrinsic race essentialism? It's very clear throughout the whole book how Sakai treats white workers as inherently racist, as if the racist elements weren't conditioned by racist ideology. In the same manner how through agitation and propaganda they can adopt a different viewpoint if an ideological force is strong enough to fight it. This is why I keep mentioning the Rainbow Coalition, which Sakai doesn't address at all, in fact the Black Panther Party is not mentioned a single time by Sakai, even though it was an extremely influent party, even after it was disbanded, and even at the time Sakai wrote the book.

The Rainbow Coalition was a successful example of racial solidarity among different ethnic groups, and avoided the co-optation of leadership by poor whites, and was only ended by the assassination of Hampton. It was so successful and threatening to white supremacist ideology that the FBI planned and executed the assassination of Hampton in the matter of 8 months after the Rainbow Coalition was founded.