If they were to genetically modify a pig to chew its cud, would it be kosher?
If the genetically-engineered pig was gestated in a normal pig, then no it would not be kosher.
Rambam in Laws of Prohibited Foods, 1:5-6 writes:
א,ה [ד] בהמה טהורה שילדה כמין בהמה טמאה--אף על פי שאינו מפריס פרסה, ולא מעלה גרה, אלא כמין סוס או חמור לכל דבר--הרי זה מותר באכילה. במה דברים אמורים, בשילדה בפניו. ...
א,ו [ה] בהמה טמאה שילדה כמין בהמה טהורה--אף על פי שהוא מפריס פרסה, ומעלה גרה, והרי הוא כמין שור לכל דבר, או כמין שה--הרי זה אסור באכילה:
If it is absolutely certain that a kosher animal gave birth to something that looks like a non-kosher animal, even if it doesn't have split hooves or chew its cud, and it looks completely like a horse or donkey -- it is kosher ... A non-kosher animal that gave birth to something kosher-looking, though it may have split hooves, chew its cud, and look just like a cow or sheep, is not kosher.
I'd assume we don't distinguish between the piglet that was born with a random mutation to chew its cud, and one genetically engineered to do so.
To quote Rabbi J. David Bleich ("The Problem of Identity in Rashi, Rambam, and the Tosafists", Tradition 41:2):
The notion of identification as a member of a species is best summed up in a pithy comment attributed to R. Chaim Soloveitchik. It is reported that R. Chaim queried: Why is a horse a horse? Is it a horse because it is a horse or is it a horse because its mother was a horse? To rephrase the question: Is a horse a horse because it manifests the characteristics that are the necessary conditions for identification as a member of the equine species or is a horse a horse because its mother was a horse? R. Chaim proceeded to declare that a horse is a horse solely because its mother was a horse and explained that ancestral identity is the sole factor that determines membership in a particular species. Thus, as spelled out by the Mishnah, Bechoros 5b, identity as a member of a clean or unclean species is determined by birth and not by distinguishing physical characteristics.
As a related question, I asked Rabbi Welcher in Queens about goats that were given genes from a spider so their milk contained gossamer, is the goat still kosher? He said that products of cross-breeding (even if doing so is prohibited) remain kosher, and as the non-kosher genes need the kosher genes to produce an organism here ("zeh v'zeh gorem"), it's allowed.
The Midrash is quoted as saying "Why is its name called chazir? Because the Holy Name Blessed Be He will return it to Israel."
A lot of commentaries learn this non literally (see here and here and here for examples). Others learn it literally. The Or Hachayim (Vayikrah 11:7), writes that when Moshiach comes the nature of the pig will change, and it will start chewing its cud. It will then have kosher simanim and become kosher.
This doesn't exactly answer your question, but perhaps when Moshiach comes the pig will be genetically modified (at least according to the commentaries that take it literaly), and be kosher.
Here's an essay based on the teachings of the Lubavitcher Rebbe, discussing why specifically the pig will become kosher again, more than any other non-kosher animal.
No, in addition to the fact that it doesn't meet the criteria for an animal that is kosher, the Torah specifically mentions pig as not allowed:
As it's stated in Deuteronomy 14:8:
וְאֶת־הַ֠חֲזִיר כִּֽי־מַפְרִ֨יס פַּרְסָ֥ה הוּא֙ וְלֹ֣א גֵרָ֔ה טָמֵ֥א ה֖וּא לָכֶ֑ם מִבְּשָׂרָם֙ לֹ֣א תֹאכֵ֔לוּ וּבְנִבְלָתָ֖ם לֹ֥א תִגָּֽעוּ׃
"And the pig, because it has a split hoof, but does not chew the cud; it is unclean for you. You shall neither eat of their flesh nor touch their carcass."
Look in Likkutei Sichos Chelek Aleph (Parshas Shmini) P. 222 where this issue is addressed.
(link can be found here)
All your answers are fundamentally flawed. First of all, what's kosher or treif is not based on looks or preconceived notions, but simanim. The OU, whom are in the hechsher business, writes:
Rabbi Belsky was asked about a salmon (a kosher fish) that was engineered with genes from an eel (non-kosher), enabling it to grow faster. Kosher fish are identified by their fins and scales. Therefore salmon is kosher, because it possesses both fins and scales. An eel is not kosher, since it does not have fins and scales. In this situation, Rabbi Belsky ruled the genetically modified salmon is kosher, since it physically resembles a salmon, and it exhibits the signs of a kosher fish i.e. it has fins andscales.
Therefore, if a pig, which already has split hooves, would chew its cud, it would pass all the simanim.
But just because it has four stomachs, doesn't make it automatically kosher like the babirusa. B’frat, it must davka regurgitate the cud vadai and have split hooves mamash. As far as the health and spiritual benefits from such genetically-modified animals, that likely retain their violent and filthy habits, that probably wouldn't change.
Another option would be lab-grown meat, in which case you could eat bacon cheese burgers. Rabbi Menachem Genack, CEO of the Orthodox Union’s kosher division, said that meat from a lab-grown hamburger could be consumed with dairy products. Since this animal is essentially soulless, this would probably obviate the deleterious health and spiritual detriments.
To answer another misconception, such an animal would not be called a pig. This is how it is explained in Toras Menachem. Even then, I doubt any chassidic or haredi Rabbi would kasher it.
However, all that being said, there is clearly an issue of maris ayin. But this can be overcome through several ways. Based on the preceding, it would appear that one would have to leave the product on the table, and the product label would clearly have to display the ingredients, hechsher, GMO status, and whether its pareve or not.
Just so you know, you can get a bacon avocado cheeseburger at Mr. Broadway in Manhatten, just don't let your Rabbi see you.
No. It would no more make the pig kosher than smashing your engine light fixes the engine.
Our Creator gave us kosher laws as a blessing, so we would know what is fit to eat and what is not fit to eat. They are not merely arbitrary rules imposed on us to make life difficult.
Therefore He gave us a way to recognize which animals are fit to eat and which are not. The two signs (for mammals) allow us to know which animals are fit to eat and which are not fit to eat.
To suggest that the cloven hoof and chewing the cud are themselves what make the animal fit to eat is the same as saying that kosher laws are arbitrary, not given for our benefit but merely a set of rules for the sake of imposing rules. This suggests a very negative view of God, which is not consistent with the Bible.