Neither "God Bless America" nor "America the Beautiful" is our national anthem. Only "The Star-Spangled Banner" is.
Things you do not need to do during the first two songs:
2. Remove your hat.
3. Put your hand over your heart. (Strictly speaking, you don't actually have to do this for the actual national anthem either.)
As it's generally the seventh inning stretch when you hear "God Bless America," you are probably already standing, but that's not because of the song. There is no reason for the PA to ask you to remove your hat. You should certainly not assume that bizarre somber and/or concussed expression that people appear to think is patriotic.
This whole thing is so weird, too. All things being equal, it's good to like your country. But rein it in, guys.