The tender lymph nodes under your jaw and near your ear are most likely reacting to the stye infection nearby, since that area drains fluid from the eyelid and surrounding skin. Small, soft, movable nodes like yours usually settle on their own within two to three weeks as the stye heals.
Next Steps
Keep using warm compresses on the stye and gently massage the tender area to help the swelling go down faster. The stiffness in the back of your neck should ease on its own as the infection settles.
Health Tips
Please see a general physician if any node feels hard or fixed in place, keeps growing, or is still there after two to three weeks, especially with fever or night sweats.