Before we start, the obvious answer is no. The game is not stated to end with the conclusion of April Fool's Day. It ends when the timer hit's zero. We do not have infinite clicks, since the game states that only accounts made before today can click the button, thus we will eventually lose. To add to that, even if the game were only to last 24 hours, we'd still be in trouble.
Plenty of people have done the math. The max time span with full coordination is around 3.4 years. That's one tap per minute at the 1 second point. This uses the peak reddit traffic statistic on the about page which is 1.79 million accounts logged in at once.
Most people are adjusting down (based on whatever metric) from that maximum figure to get anywhere from 800k to 1.2 million potential button taps.
We are currently at ~310k button presses. The game has been live for around 7.5 hours (noon est/16:00 UTC is when the announcement post was made). That's ~688 taps per minute or ~11 taps per second on average. Another redditor is recording the stats in 10min increments and our TPS has varied from 9 tps at the start of the game to a peak of ~14 between the hours of 2 to 4 pm EST. we are currently in a period of decline as the East coast of the US leaves work for home and get ready for bed and such. The current TPS is around 11 again (keep in mind these are all averages).
It is expected that the number of taps per second will continue to decline through the night. But I'm fairly sure that we'll be fine. It's going to be tomorrow around 9-10am EST where things will get interesting. At the rate we're going the count will be reaching somewhere between 800k-880k taps. My personal choice for "max participation" is around 45-50% or 804k-894k taps. Depending on just how much Europe, Asia, and Australia participate we'll fall within that spread.
Now when we get close to saturation (the point where everyone who will play has played) what you're going to see is the TPS float. By that I mean you'll see it go down to really low numbers (10 or less seconds) then bounce back. I guarantee that there are groups out there that are already planning on gathering people to watch the clock and keep it stretched out as much as possible. This behavior however is impossible to model however.
So we'll be sitting here watching the clock slowly count down, then reset at the last second. It'll be like watching the heart monitor of a dying man.
But we're not going to last forever. We're already rushing far to quickly towards the brick wall of not having any more accounts to tap the button. It's too late for us. If we had coordinated better we could have lasted a year. Maybe someone actually managed to get enough people together to stretch this out for a week or maybe two. If I had to guess, we won't see the heart monitor behavior till there's less than 1000 taps less. That gives us about 16 hours if they're really coordinated. But I doubt they'll be good enough to tap at 1 second or even close to it. So at best I think we'd get half a day extra. This is over well before the weekend though, and I'm guessing we'll be done around noon tomorrow.