Working on the principle that "correct language" is defined as the common usage among native speakers of the language: "I had a fever" is correct. I don't think I've ever seen "I had fever" in print and very rarely spoken.
In general, I think the rule is that when you are referring to something that is countable, you use an article. When you are referring to something that is not countable, you do not. Thus you would say, "I had a dog" or "I had the chair", because we can count dogs and chairs, i.e. there might be one dog, two dogs, etc. But you say, "I had food" or "I had happiness" because you cannot count "food" or "happiness", i.e. you would not normally say "two foods" or "two happinesses". Think of "a" and "the" as taking the place of the number "one".
Fever is countable. You could say "I had two fevers this year: one in January and another in March."
Which brings to my mind an interesting point: Some diseases are not considered countable. You wouldn't say, "I had two leprosies" or "I had two diabeteses". So we don't use "a" with those. I'm not quite sure what the rule is. Maybe: Long term chronic diseases are not countable, but short term diseases are? Like, "I had a fever", "I had chicken pox"a headache", "I had a cold", but "I had leprosy", "I had cancer", "I had manic-depressive disorder". Hmm, this is starting to sound like, "I have hypochondria".