The use of "man" is perfectly fine here. What I would complain is that your sentence sounds somewhat bare or naked. I would definitely insert something like "manchmal":
Man braucht manchmal eine Pause.
This way, you're saying that people sometimes need a break. Without "manchmal", the sentence sounds a bit like "Everyone needs a break right now" to me.
In spoken German, you can abbreviate "manchmal" by "mal":
Man braucht mal eine Pause.
Back to the "man": Your suggestion "Frau brauche eine Pause" is ungrammatical and not acceptable in spoken German. In written German, however, some people would use e.g.
Frau braucht auch mal eine Pause.
Even women need a break sometimes.
(The "auch" corresponds to "Even") If it doesn't occur at the start of a sentence, these people write it lowercase:
..., aber frau braucht auch mal eine Pause.
However, this is not standard German, even though the Duden lists it, saying
"besonders in feministischem Sprachgebrauch, sonst oft scherzhaft für »man«, besonders wenn [ausschließlich] Frauen gemeint sind" (source)
I'd be careful in using "frau". Since it highlights that you are referring to women, it may suggest that what you're saying does not hold for men. So "Frau braucht mal eine Pause" may to some people suggest that man don't, which may be experienced as an offense.