Herpes zoster is caused by varicella virus, the same virus that causes chickenpox. If someone with herpes zoster lesions has skin-to-skin contact between those lesions and the skin of a person who has not had chicken pox or the vaccine, that person may get chicken pox.
As it is caused by virus it is infectious diseas.

Varicella Zoster is a type of herpes virus that is commonly the cause of chickenpox. The chickenpox vaccine, often called the Varicella vaccine, is a live virus vaccine to protect against chickenpox, as it can be extremely dangerous in adults. As shingles can also result from the same virus, the vaccine protects against that as well.

Herpes zoster (or shingles) is a painful, blistering skin rash due to the zoster virus, which is the virus that causes chickenpox. What happens is that the virus remains inactive (or dormant) in certain nerves in the body and it occurs after the virus becomes active again.
It states the name of a virus and in medical terms infection caused by that virus Herpes Zoster

Yes, because genital herpes is caused by herpes zoster virus. Zovirax cream is used to prevent from herpes zoster virus from multiplying.

Herpes zoster is not a childhood disease, but the virus that can cause herpes zoster, chicken pox, is a childhood disease.