Meyers have thorns. The large thorns and larger than typical leaves on the new branches are what we call watersprouts; they usually indicate that you are being a bit too kind to your tree in terms of food and water. Not to worry, just prune them back to the general level of the canopy and they will revert to normal growth.
A mature Meyer will normally have maturing fruits, small fruits, and flowers all at the same time and will produce fruit nearly the whole year.

This is the first year that Meyer has both blossoms and fruit. I've heard they can produce all year, but I didn't think it possible in our very WARM climate!

I will most definitely prune that "watersprout". I was worried of the thorns since I don't see any on the other branches.

John has given you excellent advice, and believe me, he's got plenty of "warm" where he is, and his Meyer lemons are doing exceptionally well. Your Meyer looks extremely happy, and water sprouts are simply a sign of a very healthy and happy tree. If the sprout gets too big, simply prune as John has said. Most new and tender branches of all citrus have thorns. You should see the thorns on my Yosemite Gold mandarin!

Patty S.