Could a new country be started on Antarctica Independent of any other country?

No, there is a treaty called the Antartic Treaty, it does not recognize territories on Antartica. And prohibits military activity on Antartica. The only thing you can do there is build research stations and study the wildlife, climate and ice. But, i guess you can still make your own colony, make a sign that says "Welcome to JoshLand". You just won't be an official country. And you will die.