Each of these three words has several meanings. Break out is a phrasal verb, breakout is a noun, and break-out is an adjective. As with many other English words, the adjective can also be spelled breakout, sometimes. This is a common pattern in English. For example, throw away (verb), throwaway (noun), and throw-away/throwaway (adjective). The pattern is not universal, but is quite common.
To break out (verb) has many meanings. The original meaning was “to come out by breaking through” like a baby bird breaking out of an egg. Here are the other common meanings:
(1) To escape from a prison, jail, or any kind of confinement, or to help someone do so:
The prisoners broke out of jail in the middle of the night.
Ralph’s girlfriend broke him out of jail.
(2) To bring something “out” into a public area to share or use:
Let us break out the wine and celebrate!
(3) To develop or emerge suddenly:
Last night a fire broke out in our neighborhood.
New fighting broke out in Israel today.
Laughter broke out of the crowd.
The crowd broke out with laughter.
(4) To develop a skin problem, such as pimples, a rash, or sweat:
Chocolate makes me break out (with pimples).
As he ran, sweat broke out all over his body.
A breakout (noun) has some of the same meanings as above, and some different meanings:
(1) An escape from prison, etc.:
There was a breakout at the prison. Five prisoners escaped.
(2) A skin problem such as a rash, etc.:
Julia gets a breakout of pimples after eating chocolate.
(3) The sudden appearance of a disease in a community:
There was a breakout of monkeypox in New York.
Break-out (adjective) is less common . . . to describe . . . :
The first successful song, album, or book of an artist:
Jen’s break-out hit song sold a million copies.
A meeting where people invent and share ideas:
We held a break-out session at work today and came up with some new ideas.
These words have some other less common meaning. For example, when the price of a stock (an investment) changes suddenly, going higher or lower than it has been for some time, this is a breakout.
Finally, the verb can be used metaphorically in many creative ways:
I want to break out of my old habits.
We need to break out of our previous ways of thinking.
The verb break out comes from the Old English word ut brecan, which is over 1,000 years old!