보이차 is called puer cha in China, if i Romanized it right. It is an aged tea that is usually sold in compressed disks. You flake off the tea leaves to use them for brewing. IIRC you can buy 3 year, 5 year, and 7 year tea, and the age greatly impacts price. A lot of people in China and Korea drink it because it is believed to be good for diabetics. It is a real tea, but as i understand, it contains no caffeine.

-긴 is short for -기는. 하기는 하다 is used for emphasis. In your example, 좋아하긴 하지만 means "it IS good, but...". -지(는) 않다 is one way to negate verbs and -는 is used for emphasis here as well. So I think that means "Herb tea is indeed good, but it's not as good/tasty as 보이 tea." (I don't know what 보이 tea is.)