Learn Hardcore Portuguese (Brazil): A técnica disse que o botão foi trocado em cinco minutos. - The technician said the button was replaced in five minutes.

In this sentence, a técnica is the feminine form meaning the (female) technician.
Portuguese técnica can also mean technique, but the context (someone said something) strongly points to a person here.
If the technician were male (or unspecified but grammatically masculine), you’d use o técnico.

What do the accents in técnica mean? Do I have to write them?

Yes—accents are part of the spelling in Portuguese. Técnica has an acute accent (é) to mark stress and vowel quality. Without the accent (tecnica), it’s considered a spelling error and can confuse readers.

Why does Portuguese use disse que? Can you drop que?

Que is the normal connector meaning that after verbs of saying/thinking: disse que = said that.
In standard Portuguese, you generally don’t drop que the way English sometimes drops that.

What tense is disse?

Disse is the pretérito perfeito (simple past) of dizer (to say/tell). It refers to a completed action in the past: she said.