You are smart in asking the question. When Obama speaks, people often hear what they want to hear. He often chooses his words carefully. Such phrases as everyone has to chip in, he is using a poker metaphor referring to everyone has to pay more into the system. If we used another poker metaphor and said, "everyone has to ante up" or "everyone has to make sure the pot is right", it becomes clearer what he is getting at. He not only wants everyone to pay, he wants them to pay a larger amount.
He is not saying, "let's do this together" or "we are a team and need more teamwork", by using the term 'chip in', he is making it clear that he wants more money, and that he thinks EVERYONE needs to pay in. This is a giant poker game, with a huge pot, where the odds are in his favor. He wants to make sure that the pot is as big as possible.
Implied in his statement is that the new obamacare need 100% participation. He wants the pool of participants to be everyone in the country. His comment is directed at making sure that everyone is at the poker table. He does not want anyone to leave the game early or choose not to play. In other words, he is saying everyone will be FORCED to participate. In this case, 'EVERYONE has to chip in'.