For this particular bill, the House and the Senate would adopt rules that have been pre-defined in the Congressional Budget Act of 1974. These rules limit - with great specificity - precisely what kind of budget bills may be passed under these limited debate and amendment rules - rules that also do away with the filibuster in the Senate.
The Budget Reconciliation process is used with some frequency, but it's never used for bills courting such controversy as this health care bill does. If they want to pass health care using Budget Reconciliation, MANY deep and profound changes would need to be made to the House and Senate bills.
Before bills can be passed using Budget Reconciliation, a person(s) called the Congressional Parliamentarian will review the bill and will strike any provision that doesn't meet the guidelines set-forth in the CBA of 1974. House and Senate Parliamentarians are NOT elected members, they are officers of both chambers and are non-partisan. They won't allow any funny business because they aren't beholden to either Dems or Republicans, much like the Sergent-at-Arms.