Which algorithm can i use for predicting length of stay in coming year based on historical claims data?
If you want to do prediction using 2011 features, the answer is yes, you can do that.

However, as you don't want to use these features, the answer might be no.

Without using 2011 features, your dataset will have only 2 samples(2009 and 2010) under the assumption that every memberID is different. Prediction from two samples is neither reliable nor feasible.