Bulbs in Container

You could try putting some chicken wire over the top of your pots to keep the squirrels out?

Those bulbs are fine outside all winter. If they're in your shed you possibly could lose them to mice or damp (mould)

Even if you don't lose them with the problems already mentioned, you may find that because they have been grown inside, they are not hardy at the time you want to plant them out and they could keel over at the first frost.

Bulbs don't mind getting cold in the winter, they're meant to overwinter and can withstand severe weather, including frozen soil.

Their growth is triggered by a mixture of day length and warmth. You can persuade them to flower earlier by bringing them indoors, or just to somewhere warmer than outside - even bringing them closer to the house, where they're protected from cold winds, can make them flower earlier.

I think that that's what's likely to happen if you put these bulbs in your garage - they'll try to grow/flower early, and if you don't spot them shooting you'll also get a lot of weak, etiolated, growth and the stems won't be strong enough to support the flower heads.

If they were mine I'd cover the pots with a layer of large stones, that are too big for the squirrels to move. The bulbs will grow through them without any trouble. (All our tubs are topped with 'river washed stones' - they look quite nice.)

If you decide to use chicken wire, leave the wire in place even when they've started growing because if you try to take it off you could damage the growing shoots - unless you've made it into a sort of teepee that's well clear of the top of the pots.