Hiya!
Out of the two antennas you mention the four element quad would be better due to the higher gain.
However, you may find better results by using a cross Yagi antenna and feed the two sections 90 degrees out of phase (so as to give circular polarisation), at least that way you will not miss out on the downlink by having an incorrectly polarised antenna.
Also, unless you intend to invest in a tracking system (using an azimuth/elevation rotator), you would be better to use a simple cross Yagi, with circular polarisation, and mounted to point straight upwards (or better to point at the approximate area of the sky over which the space station will track).
Upward pointing cross dipoles have been used for years for satellite reception.
Just do a search on Google for "turnstile antenna".
73, Mark.