Learn Dutch: Zij heeft haar eigen fiets. - She has her own bicycle.

Both mean "she", but Zij is the emphasized form. You use it when you want to put special stress on the fact that she (and perhaps not someone else) has her own bicycle. If you didn't want to emphasize it, you would just use the unstressed word Ze.

Why doesn't the adjective eigen get an -e ending here?

Normally, an adjective gets an -e ending when it comes after a possessive pronoun like haar (for example: haar nieuwe fiets). However, eigen (own) is a special exception. It almost never gets an -e ending, regardless of the noun's gender or what word comes before it.

How would I change this sentence to say "He has his own bicycle"?

You would say: Hij heeft zijn eigen fiets. The subject changes to Hij (he), and the possessive pronoun changes to zijn (his). The verb heeft (has) and the phrase eigen fiets stay exactly the same.

Can I say een eigen fiets instead of haar eigen fiets?

Yes! In Dutch, it is very common to use een (a/an) with eigen, like Zij heeft een eigen fiets (literally: She has an own bicycle). This is perfectly natural and means the exact same thing as saying she has a bicycle of her own.