On flat shoes, like the Merrell Proterra's, is it possible to use gaiters (for snow)?
Gaiters can be used on any shoes... even sandals. Of course their efficacy might be limited.
The biggest thing to consider is how low the gaiters ride on the body of the shoe.
If you cinch the bottom strap as tight as you can, yet the gaiter still rides up over the top of you shoe (usually at the heel, since the front likely has a hook keeping it attached to the laces) then rocks can get in. Try a smaller size.
Small gaiters, snugged down have worked well for me in any number of sneakers / running shoes, etc.
In snow, the success rate is more limited, since even with boots, snow tends to pack in under the bottom lip of the gaiter - it just has farther to go up the heel of a boot than up the heel of a shoe.
I have some OR small/low gaiters that I use with my low and high hiking boots to keep debris out. I have some Eastern Mountain Sports heavy/tall winter gaiters for use when the snow gets over a few inches.
I have been looking for an alternative for summer. There are a number of running shoe gaiters out there. Just search Amazon.com or another website. I was also considering trying to make some using some old cotton pant legs. The answer entered by user "ppl" here had some suggestions.