Refrigeration damages the membranes inside the fruit walls, causing the tomato to lose flavor and develop a mealy texture . Not that they might not still be good fried, just not as good. Just like ripe ones stored in the fridge.

I like to fry them just after first blush when still mostly green. Seems to me the peak flavor for fried greens.

My mom cans them,slices in mason jars,little salt,pour boiling water over,seal jars,keeps well.Upon opening,empty in colander,dry on paper towels,dredge in corn meal,fries easier than fresh cut.

You can pick them at the first sign of color break.
Try to handle them gently not causing any damage/bruises.
I would let them fully air dry and store the in a box using crumbled newspaper and keep them in a cool place (NOT frig) like basement. This ways they should last for couple of weeks or longer.

I like the idea of canning. The whole point of saving the green ones is that I have too many tomatoes right now, and need to keep them awhile. I've stored green tomatoes in the fridge before for a few weeks, but I didn't know whether I was damaging their flavor/texture or not. No access to a basement, and I need them to keep as long as possible, so canning seems like the most viable option. Do they stay nice and firm with this method?

You ought to head over to the Harvest Forum where they can best advise you on the safest method for canning the green ones.
Edie

I bather that OP wants to use them fresh.