What would make breasts ooze milk?

Various, Baby, Tumor. It is very important to get this seen to asap, as sometimes a brain tumor called a pituitary adenoma is responsible, although rare, it is highly treatable early. Usually eye symptoms go along with this. Pregnancy is the most common cause, but various hormones can cause it, and other far less scary things. Important to rule those things out though, so make an appointment to see your primary care.