Is there any rule or custom not to have your hair cut by a Goy?
There is a Halachic prohibition against getting a haircut from a non-Jew. This is brought in Mishnayos (Avoda Zarah: Perek 2 Mishna 2) and in Shulchan Aruch (Yorah Deya: Siman 156).
However, one may get a haircut from a Goy if it is in public with other people present or there are passersby, or if the one getting their haircut looks in the mirror during the haircut. It is not enough to be able to see in the mirror, rather one must actually look in the mirror in a noticeable way.
However, there are Kabalistic reasons for people to be Mehader to always refrain from using a non-Jewish barber if there is a Jewish option available.
Sources:
ראה הנסמן בדרכי תשובה סי’ קנ”ו סס”ק ד’, ובכף החיים או”ח סי’ ר”ס סס”ק י”ג, מהמשנת חסידים, היפה ללב, התוצאות חיים, ועוד.
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