A hair cut is talking about cutting off lengths of the hair and with or both. Getting hair styled is using hair gels, sprays, curly irons, hair straighters to style it after the hair is cut. You can just cut your hair and not style it. It can worth either way.
The answer is hair cut. eg: "I'm going to get a hair cut" "hairs cut" would only be correct grammar if you were going to get a few strands of you hair cut...which is highly unlikely, unless you're Homer Simpson ;o)
When Ponyboy and Johnny ran away to the church. They had to cut their hair and bleach it to disguise them selves. And after Johnny cut Ponyboys, he was bout to cut his own hair , and he said no Johnny , don't cut your hair , but he did anyway