Why autogyros use pusher-, whereas, piston-engined planes use puller-configuration?
Pitcairn PCA-2 - By NASA/NACA, LARC [Public domain], via Wikimedia Commons

The original autogyros were tractor (google Cierva/Pitcairn).  

The big thing with autogyros is thrust line relative to the vertical C.G.   The pusher craze was started by Igor Bensen with his homebuilt autogyro, which was done that way to keep everything compact and minimalist, and give the pilot a nice view.  The issue with Benson machines and variants is the thrust line is above the center of mass, and if you unload the rotor disc by doing a bunt, or hit major turbulence that makes you go zero G, they have a tendency to pitch over and tumble as the prop wants to spin the machine about its vertical CG in the absence of any resisting force.  It's called Power Push Over or PPO and is always fatal.  You must NEVER go zero G in a Benson or any machine with a high thrust line.  It's actually quite similar to the mast bumping problem with teetering rotor helicopters, although that happens on the roll axis.

A lot of gyros move the engine down as much as possible to lower the trust line to the vertical C.G. or close to it, and have that awkward spindly look as a result because it means making the gear a lot longer.  The other thing that helps is a large horizontal stabilizer to resist large pitching movements and most gyros have them now.  

Tractor engine gyros have a better thrust line relationship (generally naturally lower, close to the CG) and also have a long tail with good moment arm and are generally immune to PPO (the fact that the engine is out front or behind is irrelevant in itself).  There are a couple of homebuilt gyros that use the engine out front.  You can tell however that the tractor machines just have way more structure to them so it's easy to see why the pusher setup is so much simpler and lighter.