The infinitive is the subject
To take a break twice a day is the subject of the sentence. Important is an adjective that modifies it, as a subject complement.
You are correct: an infinitive, like to take, is a noun—a verbal noun. Just like a verb, it can have direct and indirect objects, and it can have adverbs, and it can even have its own subject, but its role in the sentence is to denote a thing talked about, just like any other noun. A finite verb, like is in your sentence, asserts some claim about the nouns in the sentence. (More is in this answer.)
Since to take a break twice a day is a big noun, it makes sense to apply an adjective to it, like important. (You can't do this with a finite verb.) This sentence has the same elements—noun as subject, linking verb, and adjective as subject complement:
Taxes are important.
When the subject of a sentence is an infinitive, we often use the It is… formula and put the infinitive phrase at the end:
It is important to pay your taxes.
To parse a sentence like this, you might need to revise your understanding as you go. When you see It is [adjective], it might look like It stands for something that was said earlier. But when you see the to, you recognize the It is [adjective] to [infinitive] pattern, and you realize that the It stands for the infinitive. You know that the adjective will describe what comes next. The process is the same as when you parse a sentence like this:
It is important that you pay your taxes.
Even after encountering the to, you still might need to revise your understanding again, as in this sentence:
It is important to me to pay my taxes.
The phrase to me modifies important. The whole phrase important to me is the subject complement of to pay my taxes. The order of the words is the reverse of the usual order in English, since the subject comes last, but you will understand it easily once you are accustomed to the patterns. Or…
It is easy to understand these sentences once you are accustomed to the patterns.
In this last sentence, easy modifies the infinitive clause, and once you are accustomed to the patterns is a huge adverb that modifies is. The adverbial clause could also go at the beginning of the sentence or (awkwardly) right after easy. Even though English has very little grammatical inflection, it still has a lot of flexibility in the order of the words.
By the way, gerunds are also verbal nouns. The main difference between gerunds and infinitives is how they connect grammatically with other words in a sentence. Here's a sentence equivalent to yours, with a gerund instead of an infinitive:
Taking a break twice a day is important.
The It is… pattern doesn't work with gerunds.