Does one need to be affiliated with a university to publish papers?
Absolutely not. Affiliations do not matter; what matters is the value and the fit of the contribution to the journal or the conference to which you are submitting.
While it is not true that you should be in academia to publish papers, I'll give my two cents on what I learned so far:
If you work in industry and you write technical papers, your contributions are well-accepted, although they're typically more focused on certain practical aspects.
Authors writing by themselves are welcome, but they somehow are not very common because they lack the ability of writing an academic paper: they do not lack the message or the content of a paper, but just the structure, and that's something that you learn in an academic environment.
The reviewing process is not everywhere "double-blinded", so as a reviewer you do see the affiliation of who wrote the article, while they do not know you as a reviewer. I know that it's frowned upon, but I've read comments in the afterwards discussion (before the formal decisions) where reviewers pointed at the lack of a proper affiliation, or the "Who's this guy? where's s/he coming from?" sort of questions.
So all in all, there's no need to be an academic. But if you're not, stress enough your industry affiliation!
Technically, Maurice is right. You really don't have to be in academia to publish (even in academic journals). One famous example is Paul Erdos ( http://en.wikipedia.org/wiki/Paul_Erd%C5%91s#Career ), who I believe had no affiliation (at least in practice) for much of his life. However, most cultures have certain standard ways of doing things. Academia, and in particular the part of it that you want publish in, is no exception.
One thing you can do to help get your ideas accepted is to learn to write and talk in the language common to your research area. Specifically, find some papers in your area that you really like (even better if they are widely cited) and study how they are written. When you write your own papers, make a conscious effort to copy the writing style of the papers you like. One key part of this is to thoroughly know the relevant literature (previous work on the problem) and to mention it in your introduction and explain how your work relates to it. For each research area, there are numerous other hurdles you should jump. For example, if you're writing a math paper, do it in LaTeX. If you don't know LaTeX, learn it (ask Google, if you need help), since using it will make your paper more readily accepted. For a list of other criteria often used by mathematicians to quickly judge a paper, read http://www.scottaaronson.com/blog/?p=304 .