Our 4-H pigs are in a pen in the pasture and, yes, it gets muddy, but they do fine and love their mud hole. They have a hog hut inside the enclosure which we keep straw in when it is chilly. We set the waterer and feeder on paving blocks along the fence, so we don't have to carry anything into the pen, but can take care of them over the fence. When pigs are on dirt though they can get long roundworms, so we have wormer put into their feed. Yes, they can be on cement, but what will you do with a cement pad when you are done keeping pigs? With dirt all you'll need to do it bobcat it level one day.

Claraserena,
You have too small an enclosure is your first problem. I have kept hogs in small enclosures and it doesn't take long for a problem of bare ground and muddy conditions to develop. One of the things I can recommend you to look at is slope. If you do not have enough slope away from the pig pen it will not allow adequate drainage.
Unless you want to keep pigs in the same area, all of the time, I wouldn't recommend pouring concrete to cover the area where the hogs will be. In the winter the concrete will ice over and you can have problems with legs/hips getting broken which will mean most processing plants will not take them in for slaughter. If they do not break a leg you will run the risk of muscle strains from slips which will mean hogs not eating like they should. If your husband wants to pour a concrete pad where the feeders or waterers are, that could be of benefit, but only in that area. If you want to gravel a part of the area, that could be of benefit in that area, but will not solve the entire problem.
As to the issue of ringing thier snouts, that might be your only solution if you can not expand the size of the hog lot. But remember rooting is only one part of the problem. Each hog has four compactors, one on each corner, and each step they take tramples the grass and compacts the soil.
If you can not expand your lot or change the slope so natural drainage can be enhanced your best solution might be too expensive to think of. You might have to do like some of the Europeans have done and build a confinement building with a pit for sewage. (I have seen some that are designed for only a dozen head, but they haven't caught on here. Most confinement operations in the U.S. are hundreds of head ops.) Then you can dispose of the sewage by pumping and spraying or injecting into the soil of either permeanant pasture or crop fields during the fall and winter.

mokevinb--

Good point about putting the pen on a slope. That would have helped. I also like the idea of cementing or cement blocks user the feeder area. They have a waterer connectd to a hose and under that it is a mess, even though we have put cememt blocks underneath. We will have just three pigs, and the pen is 18 X 36.

I'd humbly suggest that a pen that small with three pigs will never be anything but dirt (if it's dry) or mud (if it's rainy) no matter how well you plant or manage the soil. In fact, a pen ten times that size would probably be barely adequate to accomplish your goal of having some solid ground.