Is "COPD (Chronic Obstructive Pulmonary Disease)" more correct, or would "COPD [Chronic Obstructive Pulmonary Disease]" be right?
Preference for the use of parentheses or brackets or some other punctuation, such as commas, would be a matter of style for a particular publication and governed by a style guide, such as the Associated Press Stylebook, the Chicago Manual of Style, the MLA Style Manual and Guide to Scholarly Publishing, or other style guides.
However, in most style guides that I'm familiar with the full name is provided on the first reference with the abbreviation in parentheses. For example:
Chronic Obstructive Pulmonary Disease (COPD)
After the first such reference in the article/document, the abbreviation can be used without need to reference the full term.
Your second example would be correct if you were quoting from some source that didn't explain the meaning of the initialism; in that case you'd be introducing words not present in the source and square brackets would be used for that.