You certainly can use an article with the other words. Here's a breakfast of tortillas, here's a filling lunch, here's I once hosted a dinner. The use of the article here is more or less the same as it is anywhere else: it indicates a specific example of something as opposed to the general concept or phenomenon.
A lot of food terms are non-count nouns or can be used in a non-count sense, for example coffee:
It's a matter of learning which words are used in a mass noun sense and which aren't. For example, breakfast, lunch, dinner, supper, brunch, coffee, and ice cream are usually mass nouns because they refer to general phenomena or materials, but it's also perfectly acceptable to refer to (for example) a brunch if you're referring to a specific one.
Sandwich is not a general phenomenon or material, but a discrete object, so it's always "I had a sandwich", never "I had sandwich." Likewise, meal is not a phenomenon or a material, but a specific thing, so it's always a meal (when it means "a specific instance of having food"...there's also a definition of meal that means "ground-up seeds", and that one is a mass noun.)
Drink is a little bit of an odd case, because it's normally a count noun if it refers to "specific things to drink" (let's have a drink or let's have drinks) but a mass noun if you mean "things to drink in general" or "alcohol" ("attitudes relating to food and drink").