In "Rikki-Tikki-Tavi," Rikki Tikki destroys Nag and Nagaina's eggs, even though he was appalled that Nag ate one of Darzee's fledglings. Are Rikki-tikki's actions justified or is he being a hypocrite?

The answer to this really depends on your own personal opinion. You can argue Rikki Tikki is hypocritical in the story "Rikki-Tikki-Tavi," but you can also argue he only attacks the snakes and their eggs in self-defense.
It is certainly possible to say Rikki Tikki is hypocritical. He asks Nag why Nag thinks he should be allowed to eat Darzee’s baby. This implies it is bad to kill the children of other animals. He then goes and kills Nag and Nagaina’s babies that are soon to hatch from their eggs, which for the animals is the moral equivalent of Nag killing Darzee’s baby. It is also hypocritical—or at least silly—to be angry at a snake for eating a baby bird since cobras are carnivores and will eat some living creature no matter what. It is silly to be mad at Nag for doing what his instinct calls him to do.
On the other hand, we can say Rikki Tikki is not being hypocritical. He does not destroy Nag’s children without provocation the way Nag eats Darzee’s child. When Nag first meets Rikki Tikki, he and his wife try to kill the mongoose in cold blood. Later, they try to kill the English family that lives in the bungalow in order to make Rikki Tikki leave. It is only after they do this that Rikki Tikki kills Nag and destroys the eggs. Therefore, we can argue that Rikki Tikki acted in self-defense while Nag did not, making their actions morally different.