Is there any current gain in a FET amplifier?
It would depend on the configuration you place it in, but the most common use case is in common-source topology. And since it's a voltage gain device with no current consumed by the input, you have infinite gain. Iout/Iin = something/0 = infinite. 

See wikipedia on common source for more on what I describe:
http://en.wikipedia.org/wiki/Common_source

The other two configurations have their own current gains. Common-drain has infinite current gain as well because the gate is still used as the input so no current can flow into the amplifier input. Common gate is different in that its current gain is just 1 because what flows into one input (the source) must be the same as what flows out of the output (the drain).