I cannot confirm anything without examination but it can be due to contact dermatitis or hives. Hives (medically known as urticaria) are red, itchy, raised areas of skin that appear in varying shapes and sizes.Mostly they are idiopathic but sometimes they are caused by viral infections(like bronchitis in your case) or medications.
Take some over the counter antihistaminics like Claritin or zyrtec to your husband.Nighttime itching can be controlled by benadryl.Also apply calamine lotion or a mild steroid cream on the itchy areas.If the symptoms persist then oral steroids may be needed. If the symptoms persist, then you can consult an immunologist. He may start with immunosuppressive drugs.
If the hives don’t go after 6 wks of treatment also, then it can be a case of chronic urticaria which needs further evaluation as it may be due to auto antibodies.Plsconsult a dermatologist in that case.
Hope it helps.Take care and pls do keep me posted in case of additional queries.