I would like to know how to evicte a family member who does not pay rent or utilities, the family member has split from his spouse and now they are trying to stick me with the bills. They are in my name. Can we disconnect?

You evict someone who has established residence the same way you evict any resident. Doesn't matter whether they are a family member or not. You have to go through the process appropriate to your local area. ANY question on law needs to include your general locale as laws vary by area.

No you cannot disconnect, that could be considered an illegal eviction and open yourself to a lawsuit.