I pronounce this with the first syllable rhyming with "
Peg". That's normally how I hear it pronounced in the USA, though it is occasionally more like "May-gun", especially when the spelling is
Maegan or
Meagan.
In my understanding the most common way to pronounce this in the USA and the UK is with the "egg" pronunciation. The "ee" pronunciation, with the first syllable rhyming with "league", is most common in Australia and New Zealand. I think in Canada both of those pronunciations are used.
I believe that the "ee" pronunciation of
Tegan was also originally Australian -- in Wales, where the name originated, it was pronounced to rhyme with "egg" like
Megan. But there was a character on the British science fiction television program "Dr. Who" between 1980 and 1985 named
Tegan. She was an Australian airline stewardess, and was named after the Australian niece of one of the show's producers, so her name was pronounced in the Australian way. Since the name was not very common yet outside of Wales when this happened, this seems to have established "Tee-gan" as the more common pronunciation of
Tegan in the USA and the UK as well as Australia, even though
Megan remained "Megg-in" in those two countries.