Mfrhtyj

Eyes is a "plural count noun" and in this case we are talking about your eyes generally, rather than specifically. As a result, you can omit the the.

See Rule #3 here:

All things or things in general: Use no article with plural count
nouns or any noncount nouns used to mean all or in general.

In the case of your other example:

I have the brown eyes

If you were talking about a pair of glass eyeballs, that could be a correct sentence, but it's referring to a specific pair of eyes, rather than a general fact about yourself.

Edit: swapping out my original examples for a better one

I have boxes

boxes is a plural count noun, so you don't need the, since that's a general statement. However if we're going to directly address the boxes, we'd say:

Which boxes? I have the boxes (over there)

If boxes is singular, you still need the article:

I have the box, or I have a box

It is incorrect to drop the article and say:

I have box

In certain contexts, the following is grammatical and idiomatic:

I have the brown eyes.

For example, if you're showing to someone a photograph of yourself as a child, and in the photo you are in the company of several children all about the same age as yourself, you might say

I have the blue eyes.

I have the bloody cheek.

I have the hat on.

to refer to the thing as a distinguishing feature. The statements above could be paraphrased as follows:

I am the one whose eyes are blue.

I am the one whose cheek is bloody

I am the one who is wearing a hat.

In the context of the photo, the feature is enough to single you out. There is only one instance of the feature in the photo.

From a comment, it appears that you're coming from a French background, so I'll answer in that context.

English article usage is similar in many ways to French usage, but not identical. One key difference is that in English we do not always use the definite article (or any article) when describing general concepts, or classes or categories.

I always have a glass of milk with dinner because I like milk not ...because I like the milk. The milk would refer only to a specific kind of milk, not milk in general.

Patience is a virtue not the patience is a virtue. Again, the patience would only refer to a specific kind of patience, such as you have the patience of a saint.

I have brown hair not I have the brown hair. Same as above.

There are cases when you can describe a general concept or class with either no article and plural, or definite article and singular. For instance:

French people love good wine or The French people love good wine

Lions are apex predators or The lion is an apex predator

There is no plural indefinite article in English, and this is a situation that would take an indefinite article.

I have a large nose.

I have small ears.

I have ten fingers.

I have a broken leg.

Why the indefinite article? Here's the OED definition of a:

Used in an indefinite noun phrase referring to something not specifically identified (and, frequently, mentioned for the first time) but treated as one of a class: one, some, any (the oneness, or indefiniteness, being implied rather than asserted).

Emphasis mine. Roughly speaking, there are lots of large noses in the world, and the speaker in my first example is stating that they have one of them.

In certain contexts, the definite article might be appropriate, as Tᴚoɯɐuo mentions: when the characteristic being referred to is the only one in the set of people under discussion, such as when looking at photograph, or identifying a person in a room.

[Looking at a photograph] Q: Which are you? A: I'm the one with the large nose.

This only works because the answerer is the only person in the photograph with a large nose.

What if the photograph had several people with large noses, and several people with red hats, but the answerer was the only person with both? Indefinite articles.

A: I'm the one with a large nose and a red hat.

And if these were plural, we would use the plural indefinite if it existed, but since it does not, we use nothing:

A: I'm the one with a large nose and brown eyes.