I have both top-setting green onions and shallots that survive through the winter in Oregon. It will depend of course where you live. Mine have made it though 9 degree weather out in the open in large pots.

I have no experience with planting the green onions from the grocery. How inventive! :D. NORMALLY what would happen is that the onions (in my case from sets) left in the ground would go to seed the next year, putting up flower clusters in the shape of a large ball. This can happen the first year, but almost always happens the second (in my experience).

So typically, people pull up their onions and eat them. If you mostly like the green part of the onions, you might order (or find a neighbor) with some top setting onions. They give you perpetual green onions and usually last many years. You pick the sets that form on the top and plant those for onions to eat. The "mother" plants continue year after year with almost no work and they also put up green onions at the base. Mine have just started doing that now.

Hope that helps some.