"Drengen kan ikke lide mågerne."

Translation: The boy does not like the seagulls.

[stikuts]

Why is the opposite of "kan godt lide" "kan ikke lide" instead of "kan ikke godt lide" or "kan godt lide ikke"?

[Xneb]

Because "kan ikke godt lide" combines a positive and a negative in a way that doesn't really make sense and "kan godt lide ikke" wouldn't be a correct word order. It's just the way the language works

[stikuts]

is "Jeg kan lide" alone a correct way of saying "I like"?

[Xneb]

Yes, there's nothing wrong with it. You're probably more likely to hear "kan godt lide", though

[stikuts]

Mange tak, that helps it make more sense.