Any quick minded Dunkin' Donut's lawyer will point out that their is a Dunkin' Donunts in office building for the House of Representatives, which is quite popular with the Members and their staff and a great many deals over legislative voting occur at this location, so there could be a claim to this statement, from a certain point of view.
Dunkin' Donuts is also the reason why most American's see the Donut as the food of choice for police (they were one of the few stores open during the graveyard shift (Midnight to 8 am) and used to give free donuts and (more importantly) coffee to on duty cops. In addition, the empty nature of the restaurant would be perfect for a police to spread out and catch up on all the paperwork he has to do (and DD saved a ton on security because few crooks are stupid enough to rob it while a cop is there).
According to Franchisechatter.com, Dunkin' Donuts is the leader and trend setter in the American Donunt Market, with nearly 60% of the Market Share and has been around since 1950 and aqquiring their biggest competitor Mr. Donut in the 1990s. Krispy Kreme is now it's closest competition, but is lagging in sales after expanding to quickly in the 00s and closing numerous stores in later part of that decade. Star Bucks and McDonalds are catching up with it, but the former has not be in the industry nearly as long as DD, and McDonald's is still pushing into the breakfast market. DD is also the only one of these restruants that is known for both Donuts and Coffee (what are you Dunkin' those Donuts into anyway?) with the sugar of the former and caffine of the latter a perfect way to wake up during the morning rush hour, with no other franchise known to excel in both fields (KK is exclusively known for Donuts, SB is known for coffee (and not thought of as "fast" given the nature of waiting in line for your name to be incorrectly) and McDonalds is more known for Breakfast Sandwiches (Which DD does as well).
Suffice to say, while you are an American who doesn't run on Dunkin', when most American's want a Donut, the majority go for Dunkin' Donuts. You might also find that the reason you don't go is DD is Primarily an East Coast chain and has little share on the West Coast (and it's still responsible for 60% of all Donut sales in the nation). In the United States, the East Coast (certainly the North East) is better known for it's more rigid work culture than the west coast, which has a reputation of being a little more relaxed. Thus the day will start early (with time zones factored in). As one of the leaders in Fast Food Breakfast in the country, this isn't an unreasonable claim.
So, while not everyone indeed runs on Dunkin' they have the stats to back up the claim that America kinda does. Comercial Speech is weird in the U.S. and while they can enforce truth in advertisement laws, Dunkin isn't making a specific claim with America Runs on Dunkin'. They don't define what they mean by "Runs" nor make any specific numerical claim (as opposed to McDonald's famous 1 Billion Served). They also don't claim what product they sell that Runs America or that it will add fuel to your day and rely on the fact that Dunkin' Donuts puts it's two biggest products in the name to inform you about the nature of what it is running. It also has facts and figures and locations to show it's not entirely a false claim and they do not claim that "All Americans run on Dunkin'" but rather enough of them do that we can say with confidence we are pretty important to a typical American's work day.
It seems you believe that Dunkin Donuts is making a "No True Scotsman Fallacy" by claiming that "You're not a real American if you don't eat our donuts and drink our coffee" OR that DD is making a claim that All Americans are fat lazy donunt munching coffee guzzling Homer Simpsons. In American Jurisprudence, it doesn't matter how you interpret the message, but rather the intent of the speech maker (DD). DD will argue that they are trying to say "Dunkin Donuts is really popular in America with peak sales occuring as Americans are on their way to work. Thus, like gasoline helps a car perform its job, Dunkin Donuts helps Americans perform their jobs. Like automobiles, not every American uses the same fuel, but the market share of cars on the road do use gasoline, not all Americans run on the same fuel, but the market share buy Dunkin Donut's".
The burden of proof that DD was intentionally being offensive rests in you and you need to show explicit language that the slogan was meant to be interpreted as coded language that would imply that "Report people who don't eat DD to Immigration and Customs Enforcement (ICE), the FBI, and the House Comittee on Un-Amarican Activities (defunct), #BuildAWallAndMakeTimHortonsPayForIt" and that DD intended it OR they meant to say "All Americans are Fat Lazy and Stupid and eat donuts and drink coffee to the point it's their only source of nourishment." which wouldn't be good when the bulk of your business is with that particular market, OR that DD is claiming that its products are "clinically proven to give gasoline like levels of energy to Americans that will give them insane superpowers that will lead to American success, Bald Eagles population spikes to such levels no American Flag will ever flutter in a breeze without one majestically soaring past and performance in the bed room with your chosen partner that will last for four hours or more without any side effects that are not awesome!" when in fact it does not cause spikes to Bald Eagle populations (or something else).
This can mean internal documents such as e-mails and meeting notes that show this intent, but the burden of proof is on you to show that this stuff exists.
The fact that the nature of the statement is not a factual claim but a simple metaphorical opinion of the company based on it's interpretation of sales data means it technically cannot be a "False" Statement. Saying "I like me and other people like me and I like this new product and if people like me, they should try it too because they might like my new product." is still an opinion. Most DD commercials are "New Sandwich is yummy. Eat this new sandwich, which comes with coffee (at participating stores). People like our stuff so much, it's almost like all this country seems to eat. The End." The only factual claim is the the deal exists if the store participates in the deal. Oh, and the picture of the product, which has been treated with so many chemicals to make it look good on the camera that it's unfit for human consumption.