This made me think that I should show them I can work hard by telling how I pulled all-nighters to finish some projects during my previous job.
I would be very hesitant to set this precedent before you even interview for the job. I don't know about you, but I do not want to go into a job with the expectation that I can be relied on to pull an all-nighter to cover someone else's mistake, be it a co-worker's or the client's. If it so happens that, yes, the sky is falling and someone absolutely has to stay at work until the issue is fixed, I want it understood that I am making a personal sacrifice of my time to do so, not that this is my typical modus operandi.
So, no, I would not mention it on the application. If it comes up during an interview, spin it to the effect that you pulled the all-nighter to cover a mistake made by someone else or to support some business-critical application that saved the client $XXX,XXX. I would also mention that I would hope that this sort of thing is not commonplace at the company you're interviewing at (I would not want to work at a company that abuses employee overtime in this manner). I might even ask if something like this were to happen, would I be able to take off the next day (with pay) as compensation.
I'm assuming you're applying for a salaried position, which many people here will argue means that you don't necessarily have a set amount of hours per day/week but that you have a job to do in a certain timeframe. Most salaried people do not get overtime pay. As far as I'm concerned, a salary is based on a 40-hour workweek*. If I was expected to pull an all-nighter, I would make it known that those hours would count against my next day's work, and I would shorten my next day commensurately.
*I know I'm going to get some downvotes for this one, but a company has to base your pay on some sort of metric. The 40-hour workweek is pretty standard (at least in the US). Any hours you work over the 40 are basically free for the company and reduce your effective hourly rate. There are plenty of studies that show that working more than 40 hours in a week does not make you more productive.