Have you considered asking for a silent appointment or a quiet chair appointment? It's rude. It's reasonable! If you find going to the hair dresser, massage therapist or nail bar difficult because of the expectation of the dreaded chit-chat, why not ask for a silent appointment. Silent appointments don't work for therapy but it could work in a variety of other settings.
As an example my daughter has a massage business and she offers silent appointments for people who don't want to talk for a variety of reasons. She has a tick box as an option for every appointment. You can find out more about her services here if you're interested. She does mobile appointments for humans and horses or you can visit her in Cunderdin in the wheatbelt of WA.
If there is no tick box for a silent appointment, you can ask for one when you book or when you arrive. If it's hard to ask verbally, consider taking a card with you like this example from Hey Jude Salon in the UK.