When I'm citing a page in a journal, and the article has an independent page numbering system, should I cite the page number of the journal or the article?
I'm not certain what you mean by an article with an independent page numbering system, but I assume you mean the following. The published journal article is, say, on pages 25-48 of the journal, but there is also a preprint available online with different formatting, and it is numbered from 1-23. Not only the page divisions, but even the number of pages differ, so there's no easy way to transfer a reference between these numberings except approximately.
The most important rule is to be clear and correct: you should leave no doubt or ambiguity about which numbering system you are using. If you cite only one version of the paper, then you should use the same numbering used there. If you cite several versions, then you need to make sure your references are clear.
The second rule is that you should focus on the "most official" version of the paper, whatever that is. In most cases, a journal paper is more official than a conference paper, which is more official than a numbered technical report in a series, which is more official than a random preprint. You want to cite the most final, complete, and authoritative version, and the one most likely to be accessible to future readers. There may occasionally be subtle cases (for example, a paper that has been reprinted in books with corrections or additions), but this generally means the journal version.
P.S. The only other interpretation of the question I can think of is for journals like Physical Review, where an article has its own page number (like 032326) and then the individual pages are given subsidiary numbers (like 032326-1 through 032326-8). In that case, if you want to cite a specific page you could refer to it as 032326-3, or maybe just 3 if the context is clear. In the bibliography, you would write 032326:1-8 or just 032326, depending on your bibliography style.