I've had a cover crop of coastal berumuda, but not voluntarily (*grin*).

One of the beauties of squarefooting is waving goodbye to the tiller every wet, miserable spring (except for those years that feature dry, rock-hard clay that fills your lungs with dust...). I guess I just skip planting and tilling the cover crop and go directly to the composted material. With a good, rich compost and frequent crop rotation, you really get a lot of the same benefits with less work. (Mel's central idea, there.)

That said, you certainly could plant an appropriate cover crop for your area in the fall and chop it out in the spring, if you like. However, especially in Zone 7 or 8, I'd use a thermal cover in the winter to keep greens growing for harvest--and toss a plastic "raincoat" over all as needed so I can keep planting spring crops while the neighbors are waiting for the soil to get friable enough to till.

I'd probably put the cover crop in the pathways, mow it, and toss the goodies on the compost heap in alternating layers with leftover shredded leaves, water lightly, and fluff as needed.

If you do a cover crop in your square-foot garden, the "soil" is so light that you can just turn it with a hand-held garden fork. That's all the tillage you need. You might have to do it a couple times to disrupt the roots of the cover crop and get it to start decomposing, but it's sooooo easy with that nice peat moss mix.

That's what I do with the green manures that I start. No tiller required!