Here in Europe there are signals on 40m most of the time but I can't speak for your part of the world. Direct conversion receivers often suffer breakthrough of AM transmissions and 40m is particularly bad for this, especially in Europe, so that may explain some of the voices you've heard. My QCX is on 30m so I don't know whether the 40m QCX is susceptible to this.
In the back of the instructions there is a fault finding section. The first thing to do is check the voltages. You can use the built-in volt meter for this, or you can use a separate meter. If you do the latter, be aware that some readings will be different as it will have a much higher impedance than the built in meter.
You should notice a difference in noise level when you plug an aerial in. For receiving any piece of wire should work - the SWR is not important.
There's also some troubleshooting tips on the QRP labs website.
Try these things then report back here.