Well, in my opinion, eye stye can not lead to black eyes. And generally speaking, black eyes are caused by injury in eyes, back, even neck. So there is nothing to do with stye. On the other hand, a stye is inflammation of the eyelid, usually inside one of the small glands lining the eyelid. And it is caused by some bacteria and allergies. When you get it, you should make sure your eyes are far from the bacteria. Also, you can just have heating a piece of garlic, which will work wonders if done right. By the way, cool tea bags can be effective too.
Stye is the infection of the eye glands at the side of eyelashes. Symptoms are common burning and pain in eyes, irritation in the eye, light sensitive. Some severe cases even lead to blurred vision and cataracts. And black eyes is most possible to be caused by injury to eyes or face. Eye stye is hard to contribute to black eyes. Plus, black eyes is not easy to be cured and sometimes blindness may also occur. So some home remedies can help you relieve it. Apply tea tree oil on blacks can reduce black under eyes. Drop some counter eye drops to decrease pain and burning in eyes. Ice compress also help with slowing down blood gather.
I am so sorry to hear that you have got eye styes, As far as I know, stye in the eyes would make you suffer a lot, such as painful eyes, blurry vision, ugly images and even lead to infections. However, generally speaking, stye would not cause black eyes, that is mainly resulted from migrane or being hit by someone else. Hope you try to get medical treatment and take care of your eye health.