The Eclipse of Darwinism was a period in which biologists doubted that Darwinian natural selection (with its heavy emphasis on gradualism) was responsible for species change. Alternative theories proposed and discussed at the time to explain speciation included orthogenesis (that there were "internal drives" to species that inclined them to evolve in one direction or another), mutationism (that species could, non-gradually, suddenly change from generation to generation), or even variations on neo-Lamarckism (which Darwin himself did not wholly rule out; his own views of heredity were rife with Lamarckian aspects). It did not involve a questioning of whether evolution of some sort was happening.

Eugenics was developed initially by Francis Galton, Darwin's cousin. Galton's own hereditary views started with Darwin's, but then modified them substantially, removing all Lamarckian aspects. Essentially, Galton was what we would today call a hard-hereditarian and a biometrician, someone who believed that Darwinism essentially was correct, though with some corrections to Darwin's own views. The biometrician school in general contained many of the earliest eugenicists, like Galton's protege, Karl Pearson, and is about as close as you can come to a Darwinian view of evolution, with the main different being their methods (which were highly mathematical and statistical).

But around 1900, something else got added to the mix: the "re-discovery" of Mendelian heredity. This initially appeared to support mutationism rather than Darwinian gradualism: heredity worked in large "jumps" and not imperceptibly tiny steps. For the next 30 years, the biometricians fought it out with the Mendelians, each with their respective evidence and interpretations, each getting somewhat further from Darwin's original program. By the 1930s and 1940s, however, it turned out that they were both right about different aspects of the complexity that is evolution and heredity, and out of all this work the modern evolutionary synthesis was forged, merging both Mendelian and statistical approaches into a robust study of population genetics.

To step back to your original question: of the models of evolution and heredity described above, eugenics is still superficially applicable in most of them. If you were a biometrician, you were trying to weed out the lower ends of the bell curve of talent, which is how Galton himself initially conceived of eugenics. If you were a Mendelian, you were trying to breed out undesirable "unit traits" — e.g., dominant or recessive traits. And so there were two different types of eugenicists out there, both arguing for similar kinds of policies. (In practice, the British geneticists tended to be biometricians, and the Americans tended to be Mendelians.) The only group that might not think eugenics was a good idea might be neo-Lamarckians; I've no idea if any of them became eugenicists or not.

Note that the debate over eugenics shifts the discussion far away from the mechanism of evolution and to the mechanism of heredity. They are obviously related but not the same discussion. This was Galton's doing and his main impact: Darwin was interested in species, Galton was interested in heredity. It is important to note that none of these people had a great understanding of heredity by modern standards: heredity was way more complicated than they grasped, and occasionally there would be some great example of this made clear to them. But that rarely stopped them from using their rudimentary understanding to justify eugenic policies, because the eugenic policies dovetailed very nicely with their other concerns (whether they be immigration, racism, classism, fear of the "feebleminded," etc.). And evolution rarely came into the eugenics discussion except in a hand-waving way.

(Ironically — or not — some of these same statistical tools came to be used to show the fallacy of some of these eugenic "solutions." Compulsory sterilization, for example, cannot effectively work against recessive traits, as the Hardy-Weinberg principle points out: if you only sterilize the few people who are showing the double-recessive characteristic, and not the many more people who are "normal" but are "carriers," then you will never change the representation in the gene pool.)

Anyway. I have not really addressed your question of anthropology — it's an even more separate discussion to be had. (There was no schism early on, but there would become one, as the Boasian school gradually exerted a stronger and anti-eugenics influence on the field.)

The key thing to keep in mind that the eclipse of Darwinism was not a rejection of the idea of evolution, but a period of uncertainty about its mechanism. Eugenics in general was fueled by an emphasis on heredity, and was compatible with the dominant hereditary theories at the time.

For a great discussion of these debates, see Peter Bowler's Evolution, or his The Mendelian Revolution. My favorite little book on the history of eugenics is Diane Paul's Controlling Human Heredity.