Usually what the airlines are hoping that you will do is to leave sooner if you are scheduled to leave within the next 48-72 hours. As a general rule, if the flight is further out than that, it won't be included in the deal, unless they are expecting multiple-day cancellations, in which case they may include more passengers so that they can accomodate folks who get stranded away from home. However, if you wait to try the change until after the airports have already closed, then the only option that you are probably going to be offered if that is your home airport is to postpone the trip until the weather clears. (If the closed airport is a connection for you, then you can try for creative routing. For instance, if you were supposed to fly from Boston to Orlando with a connection at BWI this evening, but BWI is closed, then you could ask to do something like change planes in Chicago instead. That routing would take longer, but at least it would actually fly.)
SWA will allow you to change to another flight without paying a change fee, but you still have to pay the difference in the flight cost. However, when they call a weather situation that you are eligible for, they will usually let you go on a different flight for what you have already paid. The airport in question needs to be an airport where you will have your shoes on the ground.
At this point, Saturday flights are not on the list, which means that they expect to be back to normal operations by then. So if your flight is scheduled on Saturday and your airport is on the list, then no, you are not normally eligible. However, if your airport is one where there are lots of connecting flights, and your airport is still open right now but expected to be closed tomorrow, they may be open to letting you fly TODAY so that the people who were scheduled for tomorrow can be accomodated on Saturday.