Keep your hands scrubbed, and keep your hens’ run and coop clean too, and you are unlikely to have any chicken-related health problems.
It is possible to catch illnesses from chickens. The most common of them is salmonella. This can be contracted by touching an infected chicken and not washing your hands afterwards. Chickens with salmonella also have infected eggs, so if you suspect a member of your flock may be infected do not eat the eggs.
A clean coop and run = healthy chickens
Chickens with salmonella tend to be sluggish, have poor appetite, and have slightly discoloured combs, often a shade of purple. The best way to avoid disease breaking out in your flock is to make sure they have a clean living environment. Clean out the coop regularly and make sure not to leave the same uneaten food out for several days.
It is usually factory chickens that contract salmonella, and it is uncommon for a backyard hen to catch the disease. But still health-check your birds regularly, and if you notice any strange behaviour, contact the vet.