Jblake21,
The caruncle region contains the tear duct, which is the tube that drains the tears into the nasal cavity. This is why you nose runs when you cry.
So, skin infections in that area are not that more serious than any other around the eye region.
Apply the warm compresses and keep the area clean with soap and water. Soap and water are the best antibacterial substance there are. If you want to use a little hydrogen peroxide that would be okay, but do not get it in the eye. Just plain soap and water is probably the best. It is probably best not to use any of the skin antibacterial ointments around the eye. Also, when patients use these ointments they tend to forget to wash it off and wash the red area daily. They just goop on more ointment. So, again, soap and water are the best.
If these do not get the area to improve quickly, or the redness, swelling, discomfort, drainage get worse, you probably need to see your physician. You may need some antibiotics. But, the warm compresses and cleaning are probably the most important aspects.
Good luck.