When I lived in south Louisiana, we'd hunt these critters with everything from a .22 LR to a .30-'06 to a 1911 in .45 ACP.
The .22 LR really isn't big enough, but you're shooting nutrias mostly on marshland and they can't hide. They usually give a good jump with the first round out of a .22 but come down in the same place. You know you've shot them enough times when they stop jumping (two or three shots). This is a game for the .22 autoloader.
The .30-'06 (with Accelerators) is sure death on them. My closest shot was just across a canal, maybe 40'. I was walking to a vantage point when a sudden movement startled me, and I slowly brought the rifle to bear on the previously-hidden nutria. It was so close that all I could see was blurry yellow teeth, so I held on them and squeezed. It rained nutria meat, and when I got the rifle out of recoil I could see scraps of bloody fur hanging in the undergrowth but no nutria. I figure the bullet opened up on its skull and just shredded it.
The .45 ACP with 185gr hollow-points will anchor them, and shooting running nutria with a 1911 is just about one of the finest sports around. It's right up there with hunting armadillos with a .45 auto. In fact, nutria and armadillo are great targets for defensive handguns with defensive ammo. I can tell you that ball won't anchor them.
Re eating them, I've heard from reliable sources that they are pretty good eating... but I have never worked up the nerve to enjoy a nutria barbeque myself. What they say about Cajuns, that they'll eat anything, is pretty much the truth. I've had gar balls (not testicles, balls of gar meat coated in batter and deep-fried) and they're very tasty. I've had alligator tail and it's very tasty. We all know about boudin, and it's good... just don't try and find out what's in it. One of my Cajun girlfriends tried to get me to eat ponce (cow stomach) and the mere sight of a swollen cooked stomach almost made me heave all over her family's kitchen. No thanks... I'll stick to the turkey and ham and leave the stuff you're supposed to throw out when you clean an animal alone.