No, chicken mites can't live on humans, though they will bite and leave an itchy red bump.
They can infest your house, and may do that if you have chickens or even a bird nest near you, though they will much prefer the birds.
Chicken and bird mites like to live in bedding and will bite humans while they sleep.
Bites from bird mites will be extremely itchy and irritating, though the mites themselves will not live beneath the skin of a human.
If you feel that you have chicken mites in your home, there are many sprays available that deal with this problem, though please be careful, wear a mask and make sure that the room is well ventilated and other pets are out of the way before administering the spray.
If the problem is very bad, it may be worth calling a pest control company to rid them once and for all.
Why is it when someone mentions mites, we all feel itchy..? :-)