Your first line of defense is to state that you are a US citizen. If they ask how you became a US citizen, say that you were naturalized.
The Border Patrol officer can probably look you up in the naturalization records. To arrest you, the officer needs to have probable cause to believe that you are lying about your citizenship.
The Border Patrol cannot deport you. They must put you in deportation proceedings, where you will get a hearing before an immigration judge.
To avoid carrying your passport everywhere, you could make a wallet-sized photocopy of your naturalization certificate or passport. Again, while it's not legal proof that you're a citizen, because photocopies can be forged, the Border Patrol needs to have a reasonable, articulable suspicion that it was forged to investigate further.
I am concerned that if I get stopped by border police these days, they will not accept my driver's license as proof of US citizenship.
Surely, they do not have to, because a driver's license is not proof of citizenship (even given the coding that you suspect exists in your state's licenses). But with the advent of the Real ID act, licenses are becoming a stronger indicator that the bearer has legal immigration status of some sort or another.
Can they deport me just like that (even to a country I wasn't born in) if I cannot supply documentation they're requesting at that moment?
No, see above. Before you can be deported, your claim to citizenship must be evaluated by an immigration judge. Mistakes do happen, though. Most cases I'm familiar with are not people who were naturalized, however, but who received US citizenship by automatic operation of law when their parents naturalized before their 18th birthday. Such people are US citizens who have neither a US birth certificate nor a naturalization certificate.
Furthermore, in order to deport you, a destination must be identified. So it's very unlikely that you could be deported anywhere other than your other country of citizenship (if indeed you are still a citizen of that country). If for some bizarre reason the judge found that you are not a US citizen, your naturalization could still come to light during the process of arranging for your deportation, especially if you have renounced your original nationality.
What are my rights in this situation?
The ACLU has a page on this: Your Rights in the Border Zone. In short, you don't have to answer their questions, but as a practical matter you're more likely to escape without trouble if you do.
How do I tell if they're asking (illegally) for more documentation than necessary, and how do I approach the officer if I think they're asking for more than required?
They can ask for anything; it's not illegal for them to ask. They can't require you to have any particular documentation, however. If they ask you for something you don't think you should have to show them, you can say, "I don't think I should have to show you that" and either refuse to show it or explain that you don't have it with you.
They can detain you beyond the immigration status investigation only if they have a reasonable suspicion of a crime. So once they're satisfied that you're a US citizen, they have to let you go unless they're investigating a crime. To find out whether they are investigating a crime (or are not yet convinced of your citizenship), you can ask "may I go?"
Will it be necessary, then, to always carry my passport with me at all times in the US, along with proof of employment, proof of real estate ownership, proof of not owing any income tax, etc.?
No. Furthermore, proof of employment, real estate ownership, and income tax records would be useless; any noncitizen can have these things. The Border Patrol also cannot detain you if you're delinquent in your taxes, which for the most part is not a crime.
For those that it may apply, I know that even an expired US passport IS proof of US citizenship,
Not quite; it's probably better to call it "evidence" of US citizenship. It only proves that you were a citizen when it was issued, and that you probably did not lose that citizenship before it expired. If your citizenship were revoked or renounced, the expired passport would not be cancelled, while a valid passport would be. (A naturalized US citizen can be deprived of that citizenship involuntarily only if it was obtained by fraud.)
and it's best not to travel out of and into the US when it is expired.
It's not usually possible to do that, and it violates the toothless law at 8 USC 1185(b).
However, what would one need to do if a border patrol officer insists that a naturalized US citizen is subject to deportation if that person presents an expired US passport?
A US citizen cannot be deported, period. If the officer insists on pursuing deportation, and you are unable to convince the officer of your citizenship, you will have to face an immigration judge. But really, the chance of it getting that far is negligible.