I've had this idea occur to me, too (although not in quite as much detail as you've outlined). As Mike Linksvayer mentioned, free days on public transportation don't attract many extra people. (This may be due to it being viewed as a one-time, special event--perhaps if transit were free for a long period of time--say, several months--permanent ridership might grow as people would eventually get used to going without their cars.)

In any case, I think the real gains might be seen on long-distance trains. A group of people can put up with a few extra dollars to avoid the hassle of parking or driving in downtown Boston/New York/Washington, but no family of four would seriously consider spending $1200 on Amtrak tickets when the cost of gas is far less. Even for out-of-towners, a $50-per-day rental car plus a one-way drop fee is probably a better deal than a one-way Amtrak ticket. But if Amtrak offered a family or group fare, I think many more people would consider it as a viable replacement for their cars (and if the group price were substantially lower than airfares, some families might choose it over air travel).

I doubt that anything would ever cause Amtrak to turn a profit (as far as I know, even European railways, which have consistently high load levels due to the high cost of gas and the city layout centered around public transportation), still need subsidies. But most of Amtrak's routes have abysmally low load levels and anything they can do to increase ridership--even reducing fares--would probably increase revenue. (Amtrak could still charge a premium on high-density routes like the Acela Express.)