For an incredibly in-depth look at は vs. が, I recommend Jay Rubin's "Making Sense of Japanese: What the Textbooks Don't Tell You." Much of the book is well beyond me, but that section really helped me grasp the difference. He translates many of Haruki Murakami's novels, so he knows his stuff. :D

Isn't it referring to subject, than topic

Thanks for clarifying this!

For negative sentences the particle を is often replaced by は. 「野菜を食べません」isn't ungrammatical, but は would be a bit more natural.

Edit: It does depend on the context as well, read AbunPangs answer below!

Then that's a considerably more advanced sentence. To say "I don't like eating vegetables", you could say 「野菜（やさい）を食べるのが好きじゃないです」

If you want me to explain it in detail, I don't mind, but you'll learn it later in this course I think.

Because you aren't saying 'I am not vegetables'

First a disclaimer: I'm a learner myself and much of what I'm saying is transferring feelings from Korean (which frequently works in very similar ways grammatically and also has a topic particle like は). But until a native speaker can give their two cents on the issue, maybe my feeling can still help you. With that out of the way:

I would strongly prefer は in that situation because it places the vegetables in the focus while also subtly implying a contrast to other unnamed things. I guess you could technically use を, however I would also add a demonstrative: この野菜(やさい)を食べません。 But somehow that sentence sounds very blunt to me. Like the speaker is a little child who doesn't want to eat the vegetables because they're icky. The version with は sounds a little softer to me, maybe because it implies that you will eat the rest of the food.

Also note that 野菜は食べません could also be talking about vegetables in general (although pragmatically speaking it's probably less likely). は is not a definite marker like English "the" – although there is some overlap. It simply makes whatever is marked by it the topic of the conversation, places it the focus of the discourse. That being said, making something the topic typically requires the thing in question to be something known. Typically that thing is something which has come up before (in this respect は is similar to "the") or something which you can safely assume the listener to know about even without being mentioned before (these would not be marked with "the" in English). A proper noun maybe, or a general category such as "vegetables" in "I don’t eat vegetables". For example: 野菜(やさい)はよく緑色(みどりいろ)です。 (Vegetables are usually green.) So は is an indication that we’re talking about specific vegetables but no guarantee for it.

When the kana は is used as the particle it's pronounced "wa". Check out the Tae Kim into to particles for a bit of info and a great resource to use for Japanese Grammar http://www.guidetojapanese.org/learn/grammar/particlesintro

Thats because its technically broken down as Konnichi [this day] + ha(wa)[subject]. Since its such a common phrase its been shortened. ( From an even longer sentence actually)

You can if you want to, and you would be understood by a Japanese person, but it's considered at best, charmingly redundant, or at worst, rude and self-centered.

Adding 私は makes you the topic of the conversation. In English, it might sound like someone saying "hey, we're talking about me here, and I don't eat vegetables."

-は marks the topic of conversation. Basically it places the vegetables in focus and tells you that the rest is information about the vegetables. However if whatever you mark with -は happens to be the subject or object, -は replaces the subject marker -が/the object marker -を. So if we ignore real-world knowledge for the moment, this sentence could in theory also mean “The vegetables don’t eat [it].” Real-world knowledge (and in any normal situation also context) tell you that unless we’re talking about some fantasy situation with monster food, this doesn’t make much sense and that the vegetables must be the object.

You’re right that it seems a bit more common to use -は on the object in a negative clause than in a positive one. I guess this is because one important function of the topic marked by -は is to establish a contrast, and this is quite often something you want to do in a negative clause. Think about how often people say “not x, but y”. In our case there is no overt y, but the -は hints at it existing. So when you say やさいは食べません as opposed to やさいを食べません you’re not only saying that you don’t eat the vegetables, but you’re also subtly hinting that there is something else to contrast with the vegetables. In other words, that there is something else which you do eat.

I thought the topic marker replace を when the topic is identical to the object, just like it does with が when the topic is identical to the subject? For example, if I’m asked: 野菜を食べないの？健康的な食べ物が好きじゃないのか？ I could answer like this, couldn't I: 野菜は食べませんけど、果物は大好きですよ。健康的な物が嫌いんじゃないです。 Or would you use を in the first half?