It depends what you mean. Here are three possibilities:
If you have one single egg (boiled or fried or poached, for example), you'd say 'I have AN EGG'.
If you have two or three eggs ( each one boiled, fried or poached), you'd say 'I have EGGS'.
If you have a few spoonfuls taken from a pan of scrambled eggs (where a number of eggs have been mixed up to make a creamy mass), you'd say 'I have EGG'.
If you say "We usually have egg for breakfast", you are referring to 'egg' as a substance, as amorphous 'stuff' (like soup, or snow, for example). We'd understand this to mean scrambled eggs, or something similar.
Note that the same goes for any food which comes in different forms. For example:
I had chicken with a potato
= You probably had one individual large round object (a potato) with your chicken.
I had chicken with potatoes
= You probably had three or four individual round objects (potatoes) with your chicken.
I had chicken with potato
= You probably had a mound of mashed potato with your chicken.
The relationship between the noun - singular or plural - and the article is an important one which carries meaning. There is a difference between the meaning of a noun with an article (a/an/the) and a noun without an article (zero article).