I absolutely recommend switching to a gluten-free shampoo and other hair care products.
It’s not strictly necessary, since you’re using these products on your hair(as opposed to eating them). But if you ever get shampoo in your mouth accidentally, or if you touch your hair and then put your fingers in your mouth, you risk getting glutened unless all your hair care products are gluten-free. Some people with eczema also find their skin feels better if they avoid gluten ingredients in their personal care products.
Yes, sticking with gluten-free shampoo and other hair care products could save you from several "mystery glutenings" a year . or more, depending on how sensitive you are to trace gluten.(For more on that, see my article on How Much Gluten Can Make Me Sick?)
Unfortunately, hair product manufacturers regard gluten grain ingredients as healthy for your hair, and therefore use them frequently and liberally in shampoo and conditioner.