Roy and others. Please discuss when it's right to recuse yourself. Should Kagan and Thomas have to recuse themselves? Why? Why not? Because they have opinions? I'm not sure I understand this as well as I'd like.
There are rules of judicial ethics that all judges are supposed to follow. I haven't reviewed them in years (since the time I was a judicial law clerk). Some of the big ones: recusing yourself when a family member is involved in a case, or when you have a personal / financial interest in the outcome of a case. This has led to a lot of judges divesting themselves of stock holdings, etc.
In the health care case, there are questions about Kagan because she was the Solicitor General during her time in the Obama Administration. She's allowed to have an opinion on health care policy, but if she actually provided guidance on the legality of the bill itself, it's my understanding that she should have to recuse herself. As soon as Kagan was nominated to the Supreme Court, she put up a "wall" between herself and any discussion of Obamacare. However, there are emails floating around that suggest that she attended meetings regarding the legality of the bill, which would generally be a no-no. Kagan has recused herself from something like 11 challenges to laws she worked on, but has not indicated that she feels any conflict here, and has said that she never substantively discussed health care law with the administration. Conservatives are skeptical
Regarding Thomas, from what I understand, the ethics rules don't require recusal by him. His wife is active in the Tea Party / anti-Obamacare movement, but her actions aren't imputed to him. However, a lot of people think he should step down, due to the appearance of impropriety and the unlikelihood that he would rule directly against an issue his wife is so emotionally and financially involved in. Here's some more background.
Most likely, neither recuses themselves, and a full nine Justice court hears the case. It will be interesting seeing the political bomb-throwing on both sides, though.