The word "got" is a notorious word that you should avoid at all cost. You will learn to speak and write better English without it. Though used by a considerable number of native English speakers, I personally consider all such speakers to be unwashed barbarians. I will address this matter more fully later.
The correct way of expressing a particular time when using the verb "to break" is to identify that time while using the past participle form of the verb.
Likewise for the verb "to bear:"
I was born in 1990.
He was born yesterday.
The correct use of "got"
"Got" is the past-tense form of the verb "to get." It's most common correct use is in the context of to retrieve something.
However, there is amlost always a better way to say the phrase, especially with written English. For example:
Like most verbs, "to get" should operate on a noun, not another verb. It is wholly wrong for something to "get broke" or "get broken." Either something "broke" or it "was broken."
It gained its modern slang use through its definition "to become something." Notably from the 70s in the U.S., "to get down" (generally, to dance in a modern fashion; after the disco era, to become culturally involved). While not the likely root cause of its use as a connecting verb (to get verbed e.g., "to get broken" meaning "to become broken"), it helped the problem along considerably.
And to show you how slang can continue to evolve, contemporary usage commonly drops the verb "to get." E.G., "I'm down [with that]" or "are you down [with that]?")
Therefore, I beg of you, on my hands and knees, whenever you wish to use the word "got," choose not to. Cheers.