If you have seizures from sleep deprivation, that's an ADA qualifying diagnosis. Should you request reasonable accommodations under the ADA, programs will be required to address it. A request for a maximum of 24 hours is likely to be considered completely reasonable by the courts -- in fact, a 16 hour limit is likely completely reasonable.
Any accommodation request will need to pass a few tests. It can't change the essential functions of the job -- i.e. you can't make someone else do something that is core to your job. You need to be "otherwise qualified" for the position -- i.e. with the exception of the accommodation, you need to be able to do the job adequately. And it can't cause "undue hardship" for the employer. This is the most complicated part, since exactly how do you determine "undue". The court assesses each case on it's own merits. The larger the employer is, the less effect a financial impact will have. The more residents in the program, the more the program will be required to accommodate your needs with schedule changes (since the change would be less impactful for the other residents)
You do not need to disclose this at all during the application process, nor can programs ask you. All they can ask is "Can you meet the minimum standards of this position with or without reasonable accommodations?". If you answer no, then they can refuse to hire you. If you answer yes, they are not allowed to ask what those accommodations are.
What could the program do to address your need? They could schedule you for rotations that don't involve 24 hour call, as long as the educational experience is the same. They could arrange that you split the shifts with someone else -- you each do twice as many / half as long shifts (if this burden can be spread among enough other residents). They could find a way to shorten your shifts, and then add some additional shifts to make up any missed work time -- that would be completely reasonable, even if it required more weekend shifts from you on other rotations.
What can't they do? They can't fire you over this, or not promote you because of it. To absolutely require 28 hour shifts, they would need to show that it was an "essential function" of the job, or that addressing it is an undue burden. It's clearly not essential -- especially if they have any shift based services (like the ED) or night floats they could always shorten the shift. Whether it's an undue burden is an open question. They can't ask you about it beforehand. They can't limit your career options. I don't believe they are required to write about it in any reference letter they write for you -- but I'm not certain whether they are prevented from doing so.
In any case, your best option is to not need any accommodations at all and you might want to pick a residency program whose schedule doesn't conflict with your illness. Suing your residency program in court isn't a great way to start your career. Ultimately you'll need to decide whether you volunteer to disclose this during your interviews (and risk getting dropped off rank lists, but being more certain that wherever you do match is more amenable to addressing the issue), bring it up after match but before you're hired to adjust your schedule before you run into a problem, or not bring it up at all, and only address it if a problem occurs. Choice will depend on your personal feelings on the situation, and the field you're going into.
Reading material, if you're interested:
Start with this article about the ADA and medical residency in general:
http://www.jaapl.org/content/40/4/553.full The example contained in the article is terrible -- the resident is clearly awful and using the ADA to hide behind. But it's a good summary of the law.
This is a good list of "essential functions" for medical residents:
https://med.ucf.edu/media/2013/09/A...lities-in-UCF-GME-programs-final-FEB-2015.pdf I think we can quibble about some of the bullet points, but overall it seems reasonable. Still, it would seem to exclude anyone who was deaf, which seems a problem to me (and probably to the courts).
Here is a nice legal summary of some case law with the ADA and medical providers:
https://www.healthlawyers.org/Events/Programs/Materials/Documents/AMC12/papers/P_pitts.pdf It's written in lawyer-speak, so isn't for the faint of heart.
This is another summary of medical ADA cases:
http://adagreatlakes.com/Publications/Legal_Briefs/BriefNo18_ADA_in_the_Healthcare_Setting.pdf It's not as complete as the above, but an easier read.
Disclaimer: Not a lawyer. L2D may disagree with some or all of this. If so, likely I am wrong.