Because "zip" is not an acronym, my feeling is that it should be "zip file." However, I've seen "ZIP File" and "ZIP file" commonly. What is correct?
Standard usage (dating back to its original 1993 MIME registration) is "ZIP file." Even though it's not an acronym, the use of all caps denotes that you are talking about a file in a specific format that is not to be confused with the normal English word "zip." That said, I would also be fine with "zip file" as you propose, which comes off as a little less ostentatious.
However, it is not a proper noun and I would avoid "ZIP File" (or "Zip File") entirely.
First, what would you do if it were any other type of file, e.g. an MS Word file, an image file of some type, a PDF file, etc.? Here, I think you would capitalise as in normal sentences, bearing in mind whether any word is a proper noun or not.
Secondly, are you referring to the name of the type of file (e.g. MS Word file) or to the filename extension (e.g. .docx file or .pdf file)?
If referring to the name of the file type, I would capitalise as in normal sentences, bearing in mind whether any word is a proper noun or not. Personally, I would probably use "Zip file", treating Zip as a proper noun (irrespective of whether or not it is still a proprietary name). Here I disagree with @geekahedron, in that it is still an identifying name irrespective of whether it is still a proprietary name: just as "Word file" is identifying name (even though word is, of course, a common word!).
If referring to the filename extension, I would write it in lower case (as it is normally written), possibly followed by the word "file", as in ".zip file".
Your question actually asks, "What is correct?". I suggest that there is no 'formal' or commonly agreed "correct" way of writing it, although, if you are writing it on behalf of a company or other organization, then they may dictate their chosen style.