Obviously, if $n$ is even, the regular $n$-gon satisfies this property. If $n = 3$, it is obviously impossible. If $n = 5$, suppose a pentagon satisfying this property exists. Then there would have to be a set of three sides all parallel with each other. But of the three sides, there must be two adjacent ones, so the pentagon would have two adjacent parallel sides, which is impossible. For odd $n \geq 7$, take the regular $n-3$-gon, pick a half of an edge, and adjoin a square to it. See the figure, because it's not easy to explain. [geogebra]97e344d3897642ec450edfdcb3acf0d8cdbba37d[/geogebra]