"I want to boil an egg."
Translation:Я хочу сварить яйцо.
It is in the accusative case. Only nouns ending in -а and -я get unique endings in this case. For nouns with other endings, they take the genitive form (for living nouns, such as кот) or the nominative (for inanimate nouns, such as яйцо).
Since it is a singular egg, the form is яйцо, whereas it's яйца for eggs.
So you are correct that is has taken the nominative form, but it happens that the nominative case is used for inanimate nouns in the accusative.
Probably nothing - in a more advanced course. In beginning courses, you have to learn rules, so that later you know what it is you're leaving out, when you leave it out. We're also probably learning Russian as spoken in Moscow, which is probably going to be more formal, like learning Parisian French.