My dog won't stand on his back hind leg. It's a little swollen and he whines

Can my dog take an over the counter pain medicine?

It is never advisable to give your pets over the counter or human meds without first checking with your vet. Some meds can cause very serious problems in dogs and cats.
Examine the leg carefully for any signs of obvious injury, cuts, bruising etc, be sure to check the paw pads and between toes. Ligament, or muscle injury is also likely, but you'll need to get him into the vet to have it evaluated, but don't wait too long, on the chance that it is a tendon or ligament problem having it repaired sooner rather than later will not only be less costly, but will give your dog the advantage of a more successful recovery.
On the chance that is a superficial injury a vet can best determine the proper treatment.