I received and anonymous request to discuss this, and it's also been a popular topic on Facebook since the past show, so here we go. Apparently, and I didn't see the show, there was a 'pop-up' when the Duggars were eating turkey bacon that said they don't eat pork due to religious reasons. And the request was why?
The Duggars do follow some of the Old Testament laws. Why? They feel compelled to, apparently. For instance, they do not have marital relations for seven days following her cycle or for 80 days following the birth of a girl or 40 days following the birth of a boy. However, in Judaic law, she would also have to bathe in a mikva--a ritual cleansing bath--before she could have intercourse again. Also Jews who follow these laws also don't touch each other physically as long as the wife is "unclean"--until she's bathed in a mikva. They don't hug, kiss, hold hands, or sleep in the same bed. No casual or accidental touching is allowed either. Clearly, the Duggars don't follow the law that strictly.
As to the pork--well pork is considered traif (not kosher.) There are many other foods that aren't allowed either--shellfish, scavenger fish, etc. The animal has to have cloven hooves AND chew a cud to be allowed, and fish has to have fins AND scales. There are some birds that are allowed and some that aren't. You also aren't allowed to mix meat and dairy. Clearly the Duggars mix meat and dairy together in the same meals.
The Duggars apparently aren't following all the laws of kashrut, but they've decided that pork doesn't fit into their Biblical lifestyle. I'm not sure how strict they are about it, because I can remember them eating pepperoni pizza and while you can get beef pepperoni, it's not something you find easily at a corner pizza parlour, especially in the south where pig is really popular. This could be a recent choice they've made, not one that they've followed for years. I also question the non-porkness of the hot dogs they've eaten on the show.
Now some kosher Jews aren't kosher away from home. That is, they will eat in restaurants that aren't certified kosher, will eat at friends/family's homes who don't keep kosher. They still likely won't eat the foods that aren't allowed, but it's possible the Duggars will eat pork away from home if it's given to them by friends/family or at a ball park where there's no other choice. At this point, without them clarifying those choices, we only have conjecture.
It's also possible that their health led to the choice. Jim Bob has been trying to lose weight and Michelle is, as she says, a lifelong Weight Watchers member. So it's highly likely that they saw the reasoning behind pork being banned in the Torah (Old Testament) and decided it would be healthier all around if they eliminated it from their diet. Again, it's just a possible theory.
If you have any more questions, you can search "kashrut" or "kosher laws" and will likely find better information. I hope this answers your question, anonymous. Thank you for writing in!