Two of my cats gave birth in one week. They are about 3 weeks old now, one kitten's eye is swollen shut and a whole bunch of crap comes out when I clean it. The vet clinic is far away. Can I treat him at home?if so, how? Or should I take him to the vet? He would be away from his mother for a couple of hours at least if I take him.

No, he has to go to the vet, and you can take him from his mother.
Please consider low cost spaying of the moms. I had mine done for $60 at a non profit spay clinic.

Yes, a three week old kitten csn be taken away for a couple of hours. Best to see a vet. If this is part of an upper respiratory infection, you could not treat at home.

If you are worried, bring the mom and all the kittens with you to the vet.