Someone wrote an essay about what ancaps normally consider negative rights as simply another form of an economic good. I can't remember who that person was, nor can I find the thread and the link that person posted therein to their essay.
Does anyone know what I am referring to, and if so, can you point me in the right direction?
Me.
The fallacies of intellectual communism, a compilation - On the nature of power
You are still confused. When you exchange property with someone, you must exchange your property rights to this property. That means that your property rights are prior to any exchange.
As an example of a right that involves no exchange is the right to life. You do not need to contract with every stranger you meet in order to guarantee that he will let you live. If he murders you, he will be punished for his transgression of your right to life by your protectors (assuming that you have provided for such an event).
To start with your last point first, the idea of a 'right to life' is an obvious nonsense. Your example is of 'a right to be taken care of by the criminal justice system' not of a 'right to life'. It is impossible to give someone a 'right to life' because by definition it woiuld mean that someone else is obligated to provide you with life! Life is a liberty, a freedom, it exists until transgressed regardless of whether you are on a desert island or in a state.
Your example is as equaly absurd as saying that you have a 'right to breathe' and that you do not need to contract with everyone you meet to guarantee that you can breathe. The fact is that breathing is a background condition, it is there until removed, it is a liberty. You could potentially create a 'right to breathe'; if for example you contracted with a doctor and speciified that in the event that you stop breathing for whatever reason he is obligated to keep you on a ventilator, and you have a right to be on a ventilator.
As for 'property rights' being prior to any exchange, again that is nonsense, property is a liberty. Say for example there are two men on a desert island who, because they have common sense, are cooperting. They each have their shelter, tools, and stocks of food that they have found/made/taken for themselves from nature. They have no property rights of any kind, because there is no obligation between them, they do not violate each other because it is in their mutual self interest. Yet they are perfectly capable of exchanging goods in any manner they wish.
The same thing holds for a complex modern economy. There are no property rights. If you buy a can of coke and you are walking down the street and someone steals it from your hand, can you shout after the thief 'I have a right to that property'? No, the phrase is meaningless. What you had was a liberty to use that property as you wished, which has now been removed. What a state does is provide you with a 'right to justice' that allows you to seek restitution for the violation of your liberty as provided by law. The only way in which a property right can exist is if someone is obligated to provide you with property, which would involve a rental contract, a mortgage, or something along those lines.