Learn Icelandic: Börnin eiga hvíta kanínu. - The children have a white rabbit.

In Icelandic, the word you use for "have" depends on the nature of the relationship. Eiga is used for things you own outright, like a pet (kanína), a house, or family members. Hafa is usually for abstract things (like time or an idea), and vera með is for things you are physically carrying or temporary states.

Börnin means "the children", but isn't -in usually the feminine singular ending?

It is, but -in is also the definite article suffix for neuter plural words in the nominative and accusative cases. Since barn is a neuter noun, its plural definite form is börnin. It is very common to mix this up with the feminine singular, so it is a great pattern to memorize!

Why do the dictionary forms hvítur and kanína change to hvíta kanínu?

They are in the accusative case because they are the direct object of the verb eiga. Kanína is a weak feminine noun, so its ending changes from -a to -u in the accusative. The adjective hvítur (white) must match it in gender, number, and case, making it hvíta (feminine accusative singular).

Is eiga just the infinitive form of the verb here, or is it conjugated?