How does Eurylochus convince Odyesseous‘ men to kill the oxen?

Eurylochus, one of the original Ithakians in Odysseus' crew and who presumably went through the Trojan War, functions as Odysseus' second in command even though Odysseus is skeptical of his allegiance.
In Book 12, when Odysseus and his crew have made it past Scylla and Charybdis, Eurylochus and the rest of Odysseus' crew demand that they be allowed to land on the island of Helios, the sun god, in order to rest and refresh themselves. Odysseus at first refuses, but after a plea from Eurylochus, he agrees that they can rest on Helios, as long as everyone--including Eurylochus--swears an oath not to touch cattle that are sacred to Helios.
After a month of bad weather, stranding the Odysseus and his crew on the island, they have run out of provisions provided by Circe. Standing near the ship is a herd of Helios' sacred cattle, and Eurylochus, while Odysseus is asleep, tells the rest of the crew that they should offer the proper sacrifice of the cattle to the gods and then eat what they need:

And if he [Helios] is angry at the death of his cattle and chooses to wreak our ship . . . I, for one, would much rather drown all at once and be swallowed up by the sea than keep starving to death. . . . (12:326-330)

His argument is that when they return to Ithaca they can make amends by building a temple to Helios and making appropriate offerings. Eurylochus, of course, is assuming that the gods are reasonable, and he should know by now that they are manifestly unreasonable and vengeful. When Odysseus returns from his sleep, he is outraged by his crew's sacrilege but cannot undo the damage.
When, after the crew eats more cattle for six days, a favorable wind allows Odysseus and his crew to leave the island, Zeus, at the request of Helios, sends a thunderbolt crashing onto Odysseus' ship, drowning all the crew except Odysseus. Eurylochus, then, becomes the instrument by which all of Odysseus' remaining crew dies, and his wish--that he drown rather than starve--is fulfilled.