If the only thing the material did was "reflect", and it was like a flag on a vertical antenna, then I guess it wouldn't have any effect on the radiation pattern.
The problem is that no such material exists.
One problem is that for a material to "reflect", it has to be many (at least 10, say) wavelengths long. Any less than this, and effects like diffraction will be significant, and so the material won't behave very much like an ideal reflector. There are "reflectors" that are less than many wavelengths long, such as the elements of a Yagi antenna, but these work by being tuned to very specific sizes, and that's probably not what you had in mind.
Another problem is that any material that you might consider for this purpose (like metal), will have conductivity and permeability significantly different than that of air. The electromagnetic radiation an antenna produces is the result of mutually perpendicular electric ($\vec E$) and magnetic ($\vec B$) fields. In the case of a vertical antenna, the electric field is vertical, and the magnetic field horizontal, concentric with the antenna:
"Felder um Dipol" by Averse - http://de.wikipedia.org/wiki/Datei:Felder_um_Dipol.jpg. Licensed under CC BY-SA 3.0 via Wikimedia Commons.
Were you to put a metal "flag" on this antenna, it would be coplanar with the electric field. Metals are very conductive (that's what gives them their shiny appearance), and so they will have a very significant effect on that electric field, which in turn affects the radiation pattern.
Put more generally, your line of thought seems to be based on reasoning that works with mirrors and visible light. But even though visible light and radio waves are electromagnetic radiation which differ only in wavelength, this turns out to be very significant. Visible light has a wavelength on the order of 500 nm, whereas the radiation from a 2.4 GHz Wi-Fi has a wavelength of approximately 125,000,000 nm, or approximately 250 thousand times bigger. If you could make your bathroom mirror 250 thousand times smaller you'd find intuition breaks down, and you will start seeing effects not usually seen in everyday life simply because they are so small.