Baby America refers to early Colonial America which was when America was under British rule. The American colonies were one of the most well treated colonies of Britain and America wanted to be treated better, etc.
When a person is referred to as "being a baby America" it means they have an adequate amount of something and complain and want more.
-Ugh, I so don't want to go to class today. -Isn't this your only class this week? Dude, you're such a baby America