TL;DR: The history is passable accurate. The crabs are thrown back because they're too small, not because they looked like dead samurai. So, folklore aside, the battle did not cause the crabs to evolve to have those patterns.
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The history is accurate I suppose. I can give Sagan a pass for not going into the complexity of the politics of the time. The Taira clan, sometimes known by their other name, the Heike (which means "Taira clan"), as Sagan calls them the, had been the strongest samurai clan in Japan since the conclusion of the Hōgen rebellion in 1156 and Heiji rebellion in 1160. Since 1180 they had been fighting the Genpei war with the Minamoto clan, or the Genji (who they also fought in the Hōgen and Heiji). Although neither had claims to the throne, unlike what Sagan said, but they were indeed fighting for the place of the most powerful clan in Japan. Emperor Antoku had a Taira mother, a common way of securing power and influence by marrying your daughter to an emperor to sire the next emperor with your blood. By 1185, though, the Taira were pretty done, and Dan-no-ura was their "last stand" in the Shimonoseki straights. And yes a lot of them did throw themselves overboard when the battle was lost.
The Minamoto clan's victory lead to the formation of the first samurai government, the Kamakura Shogunate.
As for the crab. I'm no biologist, but:
"Interesting reading, but it isn't true.
The grooves and ridges on the backs of crabs have specific purposes and are not merely decorative. The grooves are external indications of supportive ridges, called apodemes, inside the crab's carapace that serve as sites for muscle attachment. Elevated areas between these grooves allow for an increase in internal space, so that the various parts of a crab's viscera — gastric, hepatic, cardiac, branchial, etc. — are reflected externally. This is not to say that these structures are unaffected by selection. They are as subject to evolutionary pressures as any other feature of a crab. The point here is that these ridges and grooves occur in nearly all members
of the crab family Dorippidae, whether they live near Japan or not. As pointed out by the great Japanese carcinologist Tune Sakai, there are at least 17 different species of crabs in two families in the
Indo-West Pacific that are similar enough to be called Heike-gani by local residents, and there are many related species from other far off waters that bear a likeness to a human face. Many Asian countries have vernacular names to account for the similarity
of such crabs to a human face, such as the Chinese name Kuei Lien Hsieh (Ghost or Demon faced crab), and in several countries the crabs play a prominent role in local folklore, sometimes being
considered sacred, with the face representing that of a deceased relative. In the northeast Atlantic, the crab Corystes cassivelaunus, although only distantly related to dorippids (and belonging to a separate family, the Corystidae), bears a similar arrangement of grooves and elevations of the carapace that have resulted in one of its common names, the "masked crab." Additionally, fossils of dorippid crabs or closely related crab species are known from sites predating man's appearance on earth. Furthermore, and most damning to the myth of reincarnated samurai warriors, the fisherman who make their living from the Sea of Japan do not eat any of these crabs. Whether they resemble a samurai, a human face, or merely a crab is a moot point; all are thrown back. For Dorippe japonica reaches a maximum size of only 31 mm (1.2 inches) across the back, not at all worth the trouble of retrieving from the nets, let alone sorting through to see which ones resemble a face and which do not.
So no, Sagan's probably wrong on this one, too.