There may be two reasons: First, the water temperature is higher than the temperature of your skin when you have a shower, it can increase the blood which is allocated to the surface of the skin. It can not only make you get bloodshot eyes, but also make your skin bloodshot.
Second, all kinds of shampoo, cleansing products can give a physical stimulation to your eyes, which can cause inflammatory reactions. You can use some topical antibiotic eye drops or ointment according to the severity of the symptoms. But hypertension, cerebral thrombosis can also make you have bloodshot eyes, so for safety's sake, you can go to the hospital to have a careful examination. Hope these suggestions can help you.