Q: Do ticks bite chickens?
A: Probably not. Chickens actually eat deer ticks, wood ticks, and dog ticks (as well as other types), and these are the most common ticks that bother humans and canines. Perhaps these ticks are what you are seeing? If so, there is probably no special cause for concern.
Even when you live in Tick Central, where a half hour hike means you may have to brush off many, many ticks from your clothing, you'll have few ticks in the immediate area of the chicken coop because the chickens will eat all they can find. This is especially true if you keep a breed that's a good forager.
There is a "fowl tick," Argas persicus, (sometimes known as "blue bug") that can sometimes be a problem for chickens and wild birds, but it would not likely be the type you are seeing on you or your dogs. The small Lone Star tick will sometimes attack poultry, too.
The bottom line is that if you are concerned about ticks on your flock, you can keep an eye out and examine your birds every so often to make sure the ticks in your area are the prey, rather than your chickens. You might also want to contact your local extension agent to see if there are any ticks in your area that are known to attack chickens or other fowl.