I am a native U.S. speaker, and either sounds perfectly correct.
I'm going to see what a Google Books search shows me, regarding the forms with and without the "s," by searching on the catchphrase "Take two aspirin(s) and call me in the morning," the reply doctors supposedly gave if you called them in the evening with something unimportant.
Book title: "Take Two Aspirins, But Don't Call Me in the Morning"
“Take two aspirins and call me in the morning. You'll feel much better.”--2007 book on "Applying Helping Skills"
"Even when your doctor responds to your night time phone call by saying 'Take two aspirins and call me in the morning,' he or she implicitly invites a follow-up if things aren't going well."--Psychiatry on Trial: Fact and Fantasy in the Courtroom
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''Take two aspirin and call me in the morning,' may not be as innocuous as it sounds"--1985, Vegetarian Times
"Kelvey's solution was like the old medical dogma of 'take two aspirin and call me in the morning'--2010 novel
"The advice to 'Take two aspirin and call me in the morning' has been repeated so often that most people think one aspirin is only half a dose."--Arthritis: What Really Works, 2012
But I don't know WHY both are correct! I think the reason why is that "aspirin" can be either a mass noun ("a bottle of aspirin," "81 mg. of aspirin," "anti-coagulant effect of aspirin") or a regular countable noun. In the countable noun I think it is mentally an abbreviation for "aspirin tablet."
The following sentence is awkwardly worded but if I say it aloud to myself, it makes sense and sounds to me like correct grammar and usage:
"At Costco, a bottle of aspirin costs $X, but it's too large for my family; we would never use 500 aspirins before the expiration date."