Hi Fred,
Yes you can remove them by hand but it is always better to use a winder when possible. Yes you do run the chance of distorting the spring into a cone shape when removing by hand. This results in power loss as the coned spring applies pressure and drags on the bottom and cap of the barrel.
You mentioned that your winder resembles the Bergeon Mainspring Winder. Does yours have the sleeves with it? If so you can still remove the springs with this winder. The Bergeon doesn’t have the tail-stock that the Ollie Baker has so you just need to hold the barrel with a gloved hand while winding the spring enough to slip the sleeve over the spring and into the barrel. You can then unhook the spring from the barrel and slip it off of the captured spring. If you don’t have the sleeves and would like some instructions on removing by hand please let me know.
Disclaimer: Always wear gloves and eye protection when working with mainsprings and playful kittens. 
Bob