Yes - cats have their own herpesvirus, just as irritating as the human variety, and liable to flare up with stress just like it does in human sufferers. However cat herpesvirus mostly affects the eyes and respiratory system.
The great majority of cats suffer their first bout with herpesvirus when they are kittens. They get watery eyes, sneeze, snuffle, go off their food for a day or two and look very sad and fluey. Antibiotics to prevent secondary infections and good nursing pull nearly all of them through.
Although most infected cats then become carriers very few have any further trouble with the virus themselves.
A few poor cats go on to develop flu-like signs or various eye diseases whenever they are stressed.
Some just get watery eyes. Painful ulcers on the cornea, the window of the eye, and on the conjunctiva, make other cats very miserable. Occasionally a part of the cornea dies and turns black, or the eye perforates.
Any eye problem is an emergency. If your cat has watery or pussy eyes, is squeezing his eyelids closed, or the eye looks blue, red or black bring him straight in to the surgery.