Solution 1:

There are several meanings for the word “sentence”. The online Merriam-Webster describes “sentence” as “a group of words that expresses a statement, question, command, or wish”, without restricting its grammatical form. But it also describes “sentence fragment” as “a group of words that is written out as a sentence but that lacks a subject or verb” (as the short version of the definition), thereby implying that a sentence proper has both a subject and a verb (predicate).

In a more grammatical approach, it is the presence of a predicate that makes a sentence. Many languages allow the omission of a subject, since the agent is expressed by the inflected form of the predicate. Even though this might be described so that the subject is present, but implicit, there are also many languages where sentences with no subject are normal, e.g. sentences corresponding to “It is raining” do not require a formal subject.

It is still a matter of definition what you call a sentence. And it is anyway a fact that people frequently use expressions that are not sentences in the grammatical meaning mentioned above (i.e. lack a predicate), yet serve the function of such a sentence. They are very common in short notices, news headings, etc. Usually the predicate can easily be inferred, as regards to the meaning, though we might have different ideas of the exact wording. But the point is that such wording does not matter.

“System shutdown in 60 seconds” can be read as “System shutdown will take place in 60 seconds” or in a few other ways, but the meaning is the same. Omission of the predicate probably makes communication more efficient here: the message is shorter and can thus be read and grasped faster.

However, such expressions are not accepted in normal flow of written nonfiction prose, or in “formal prose” as some people might say. The reason is that such forms of language have been developed to have a certain structure, including “complete sentences”, i.e. sentences in the grammatical sense. This makes them more readable, since this is the form we are used to expect. When you use sentence fragments, you break the normal flow. This may be an important technique in narrative text and even in nonfiction prose at times. But when used just because you don’t care to formulate your messages as normal prose, it makes the impression of… lack of care.

This is why Microsoft Office Word shows an error message in Spelling & Grammar checks, if you try to use such expressions. This can be changed by editing the settings. The way to do this depends on Word version, but the setting in Proofing options is “Fragments and Run-ons”. Even if this setting is checked (as it normally should), Word accepts fragments in many contexts. For example, if you write “Hello world” as a heading, or as paragraph of its own, it will pass. We can say that Word expects it to be a title or something like that and checks just the spelling, not the grammar. But if you end with a period, Word interprets that it is meant to be a sentence and says: “Fragment (consider revising)”.

Solution 2:

I think the correct formulation may be "The system will shutdown in 60 seconds"

A shutdown or system shutdown is a thing, a term from (mostly computer) engineering. I believe the phrase you have quoted would expand to

A system shutdown will occur in 60 seconds

or

There will be a system shutdown in 60 seconds.

The sentence as you have quoted it is a nominal or nonverbal sentence. It still carries all the necessary meaning, since the implicit verb (some variant of "to be") is clear to any native English speaker. While nominal sentences are often described as informal, that is not entirely appropriate in this case since "System shutdown in 60 seconds" sounds like a recorded or printed message which would be played or displayed automatically. So it is not a casual way of phrasing. It is impersonal, which is often the case with automated warnings and notifications (such as the "large vehicle reversing" warnings broadcast by modern heavy goods vehicles).

I would happily call such a sentence irregular but I do not find it useful to call it a fragment, which seems to imply that more should be added to the sentence before it can fully serve a purpose or convey all the intended meaning. That really is not the case here. The impersonal structure emphasises the sense of alarm and urgent necessity for corresponding action. The sentence expresses what it means. It starts and stops where it should. It is a sentence.

My question: am I wrong? am I missing something?

You are not wrong. You are not missing anything. The autocorrect function on your word-processor is missing something, because it is not sophisticated enough to detect style. It is some time since I used Microsoft Word, but its autocorrect function often used to hi-light both that and which, warning that possibly the other one should be used. It was (and probably still is) too stupid to be able to tell which of those two words was appropriate.

Solution 3:

"System shutdown in 60 seconds" is an incomplete sentence, or sentence fragment. It lacks a verb, and it lacks punctuation. It gets the point across, but it's not a whole sentence.

An incomplete sentence, or sentence fragment, is a set of words which does not form a complete sentence, either because it does not express a complete thought1 or because it lacks some obligatory grammatical element, such as a subject or a verb.2