I have been putting off writing this post for a long time. Partly because it’s a hell of a bummer of a topic and partly because I was still doing research. The research is done and it’s time to face the facts: we have to abandon the vegetable garden.
Why? The short answer is that the ground is contaminated with the eggs of Ascaris worms from the pigs. We are ideal hosts for them, as are the chickens and the Boll Weevil cat. Parasites like these are why you are warned against composting dog and cat feces.
What we have learned here is that pig farming and gardening should never ever be combined. This is common knowledge in tropical and third-world areas, but totally unknown to us temperate North Americans.
Here’s the long answer:
The latest pigs had large roundworms. This is pretty common in modern domestic pigs. The breeder we use usually worms the piglets before we get them. Either one or both of our piglets evaded him on worming day or we already had Ascaris ova in our soil (from previous pigs or passing cats, who love to use the garden as a litter box). We wormed them and thought nothing more of it, as it did not affect their meat.
In pigs, roundworms are called Ascaris suum and in humans they are Ascaris lumbricoides. However, these worms are “indistinguishable morphologically”  and are now theorized (though not proven) to be the same species. The ability of A. suum to infect humans was proven last year in a case study by the CDC. 
Ascaris eggs cannot be killed by freezing temperatures. Bleach cannot kill them, either. This means that there is no chemical we could drench the soil with that would not also kill us and the wildlife. They often survive microwaving and irradiation. They have been repeatedly proven to live 5 years in the soil – even after desiccation – and one study I read showed that the eggs were still viable after 15 years.
Temperatures of 122 degrees for a minimum of 2 hours can kill them. Soil solarization can (but does not always) raise the temperature of the soil to 140 degrees. However, it only affects the top 6 inches of soil and we have eggs at least 2 feet down thanks to the holes the pigs dug – and likely rainwater and earthworms have carried them farther down yet. 
So we don’t really have any realistic options for removing the eggs from our soil. (An unrealistic option would be to excavate the site and replace the soil entirely. However, that would be massively cost-prohibitive and unless we sent the soil to a hazardous materials dump site – which is even more costly – we would be passing the problem along to someone else because any facility that accepted the soil would screen it and resell it as fill to another customer.)
I am not convinced that microscopic ova cannot leach upwards into clean raised bed soil, but I might consider installing raised beds to grow ornamentals or plants whose edible parts have no contact with the soil, though this would limit our options to peas and corn. Anything else sits in or on the ground. Matt has suggested his pet crop, wheat. I think this might be a viable option so long as we are careful during planting and harvest about dust (eggs can become airborne and be breathed in) and do not use the straw in the chicken coop.
Our best bet is to move the food growing operation to the front yard and hope that the ground there has not already been contaminated via compost, my boots, and my tools. Over the former vegetable garden we would be best off to grow grass or “pave” the area by putting in hardscaping and/or a structure such as a pergola or greenhouse.
Please learn from our mistake. Keep pigs well away from sites where edibles are being grown and do not use pig manure to fertilize crops. Rotating pigs and vegetables, as we have done for the past several years, is a terrible idea.
 Miller, Leigh Ann, et al. “Cross-Transmission of Ascaris Infection from Pigs to Humans at an Organic Farm-Coastal Maine, 2012.” Center for Disease Control and Prevention. (https://cste.confex.com/cste/2013/webprogram/Paper1636.html)
 Leles D, Gardner SL, Reinhard K, Iñiguez A, Araujo A. “Are Ascaris lumbricoides and Ascaris suum a single species?” US National Library of Medicine National Institutes of Health (http://www.ncbi.nlm.nih.gov/pubmed/22348306 )
 Selendy, Janine M. H., ed. Water and Sanitation Related Diseases and the Environment: Challenges, Interventions, and Preventative Measures. John Wiley & Sons, Oct 7, 2011.
 Brownell, Sarah A. and Nelson, Kara L. “Inactivation of Single-Celled Ascaris suum Eggs by Low-Pressure UV Radiation” Appl. Environ. Microbiol. March 2006 vol. 72 no. 3 2178-2184 (http://aem.asm.org/content/72/3/2178 ) [NOTE: This is just one of many scientific articles I read in which sodium hypochlorite is mentioned as being used to remove the outer layers of the eggs to hasten embryonation.)
 “Standards for the Growing, Harvesting, Packing, and Holding of Produce for Human Consumption; Current Good Manufacturing Practice and Hazard Analysis and Risk-Based Preventive Controls for Human Food; Draft Qualitative Risk Assessment of Risk of Activity/Food Combinations for Activities (Outside the Farm Definition) Conducted in a Facility Co-Located on a Farm; Availability; Proposed Rules.” Department of Health and Human Services Food and Drug Administration 21 CFR Parts 1, 16, 106, Et al. (http://www.gpo.gov/fdsys/pkg/FR-2013-01-16/pdf/2013-00123.pdf )
 “Bio-intensive Approach to Small-scale Household Food Production.” International Institute for Rural Reconstruction. (http://collections.infocollections.org/ukedu/uk/d/Jii06be/3.5.html )
 Stapleton, J.J.; Wilen, C.A.; Molinar, R. H. “Soil Solarization or Gardens & Landscapes: Integrated Pest Management for Home Gardeners and Landscape Professionals” UC Statewide IPM Program,
University of California, Davis. http://ucanr.edu/sites/Solarization/files/114635.pdf )
 Hotez, M.D., Ph.D., et al. “Soil Transmitted Helminth Infections: The Nature, Causes and Burden of the Condition” Disease Control Priorities Project. (http://www.dcp2.org/file/19/ )
 Kraglund, H O; Grønvold, J; Roepstorff, A; Rawat, H. “Interactions between the nematode parasite of pigs, Ascarissuum, and the earthworm Aporrectodea longa.” Acta Vet Scand. 1998;39(4):453-60. (http://www.ncbi.nlm.nih.gov/pubmed/9926459)
 Weintraub, N.D., Skye. The Parasite Menace: A Complete Guide to the Prevention, Treatment and Elimination of Parasitic Infection. Woodland Publishing.