I have heard people say that a 6 piece yarmulke is preferable to a 4 piece one. Is there any basis for this?
Well, when the kippah is made of four parts, it forms a "tzeilem" (a cross) on top and some people are very careful to not make a cross, because it represents years of persecution and it is also a symbol of avodah zarah. That is also why some prefer a single-daled knot on their tefillin shel rosh as opposed to the double-daled knot, which is normal Ashkenazi minhag.
I'm sure there are other reasons, such as a six-part kippah fits better on the head than a four-part one, but this is one of the reasons I've heard people bring up when they say they like a six-part one.
Take a plain piece of fabric, cut it in a circle, and try to get it to lay flat on your head. Unless you're super-flat-headed, it won't stay! Thus, many yarmulkas are made of multiple sewn parts -- they contour better on a round surface; making them more comfortable and more likely to stay on.
Those sewn of six parts tend to stay on a round head better than those sewn of four parts. (Source: personal experience ... plus some common sense.) So I'd see why people would prefer them; that's really all there is to this, to the best of my knowledge.
Personal preference and style. The reason to wear a yarmulke invokes no need to have any particular style. There are those who have associated yarmulke styles with belonging to certain communities but this has nothing to do with the intent of wearing a yarmulke to begin with and is only a social construct created after the custom of wearing a yarmulke became universally accepted by Jews.
To show that the reason for wearing a yarmulke does not indicate whether one should have one made up of 4 panels, 6 panels, or any number of panels this is the reason for wearing a yarmulke: A yarmulke is made up of the words yar, mul, and ka (Fear the regime of Hashem). This is to say a Jew who wears a yarmulke is expressing the desire that they should always have the fear of G-d with them wherever he goes. This means that he fears feeling distant from G-d as a result of negative actions which he then won't take. One is reminded of this fear of G-d by wearing a yarmulke always.