MorganS:
Please draw a schematic. A bad pencil sketch photographed with a phone is better than trying to use words.I suspect that you will need a PNP transistor to switch the positive power to the GPS. Switching the negative is a bad idea as it's still connected to the Arduino inputs.
To be honest I didn't know how to upload a screenshot. Now you can check for the attachment on my initial post. :-[
Putting both Arduino and ublox to sleep means that I am looking for an energy-sufficient project. That's why I use a NPN, otherwise I will need all the time to have the PNP transistor connected to VCC.
The problem is with npn-sw.jpg where Arduino never sleeps/or wakes up endlessly. No button pushed!
Why when I just remove/disconnect the switch, like npn-ok.jpg everything work fine? This is what I can't understand.