First, what is the scriptural basis? There are two instances where the Tora lists the signs kosher land animals have to have: split hooves and chewing the cud. After stating this, the Tora says again that animals that have only one of these signs are nonkosher and lists four animals: camel [gamal], hyrax/rock-badger [shafan], hare [arnevet] and swine/pig [chazir].
Nevertheless these shall ye not eat of them that only chew the cud, or of them that only part the hoof: the camel, because he cheweth the cud but parteth not the hoof, he is unclean unto you. And the rock-badger, because he cheweth the cud but parteth not the hoof, he is unclean unto you. And the hare, because she cheweth the cud but parteth not the hoof, she is unclean unto you. And the swine, because he parteth the hoof, and is cloven-footed, but cheweth not the cud, he is unclean unto you.
Nevertheless these ye shall not eat of them that only chew the cud, or of them that only have the hoof cloven: the camel, and the hare, and the rock-badger, because they chew the cud but part not the hoof, they are unclean unto you; and the swine, because he parteth the hoof but cheweth not the cud, he is unclean unto you; of their flesh ye shall not eat, and their carcasses ye shall not touch.
Note first that the plain Tora text never makes any claim that this list of four animals (gamal, shafan, arnevet and chazir) is exhaustive. This claim is only made later by the rabbis:
The Ruler of the universe knows that there is no other beast that chews the cud and is unclean except the camel [and the other animals listed]; […] The Ruler of the universe knows that there is no other beast that parts the hoof and is unclean except the swine; […]
(Talmud, Chullin 59a)
Unfortunately, there are many problems with this claim. It is not certain to which species exactly the animal names refer to, especially in the case of shafan and arnevet (e.g., R. Natan Slifkin: There Are No Kangaroos In Tehillim). The hyrax and the hare seem to be the most likely candidates. But both animals do not chew their cud in the traditional sense of the word. If we include what they do in the definition of chewing the cud, we would have to include many other animals (e.g., Rabbi Natan Slifkin The Torah Encyclopedia of the Animal Kingdom, Chapter on Hyrax). And there are other animals more similar to the camel that chew their cud like the llama (e.g., Rabbi Yaakov Menken: Shemini).
I am not a zoologist and know little about animals, but I have tried to find out what other animals have only one of the kosher signs. It turns out to be a surprisingly difficult task, but here is my attempt. I have found some animals that might belong into the same category than the camel, i.e., ruminant without split hooves (actually Wikipedia lists all of them as pseudo-ruminants with 3-chambered stomaches instead of 4-chambered, just like the camel), I’ve found the llama, alpaca, vicuña, guanaco, dromedar, bactrian camel and the kangaroo. If we broaden the definition of ruminant to include ruminant-like chewing or similar (as for the hyrax and the hare), koala, proboscis monkey, capybara and rabbit have been proposed as possible candidates. As for the last category, animals with split hooves that don’t chew the cud, there are warthog, hippopotamus, aardvark and peccary.
Some links for further reading:
Natan Slifkin: The Camel, the Hare and the Hyrax (there is also a book by that name, this is a link to a blog article with a summary)
Ultimate ungulate, Your Guide to the World’s Hoofed Mammals
Alter Cocker: Proof of God from Kosher Animals
David Goldstein: Of Hare and Hyrax, of Torah and Science