You must choose to rasterize your output, it's as simple as that. Please read the online help on what features SWF actually supports as native vectors. Shape layers are not among them!
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As Mylenium says, you have selected Ignore for the setting that tells After Effects what to do with unsupported features (i.e., features that can't be exprted as vectors in SWF files). Shape layers are not among the supported features, so you have told After Effects to ignore them.
See this page.
I gave the a try and noticed that the color quality of the other elements in my banner. (colored text) had poor .jpg quality even when I upped the .jpg quality. Do you have any recommendations on how I can rasterize my border? Ill jump into illustrator or photoshop and play around with some files and them import them to see if I can find a work around...