Translation:We were born in this hospital.
"wurden" is past tense of "werden" which is used to express passive tense, so yes, the sentence is correct.
"waren" is the past tense of "sein" which is used with adjectives and nouns. You're probably thinking "waren" because some past participles double as adjectives, like for example "surprised."
"We were surprised" - "Wir wurden überrascht." - "Wir waren überrascht."
It can mean the state of being surprised at someting, or the (passive) action of being surprised by something. Sneaky to translate since you don't differentiate between the two in English. But that's it. Because the Duo sentence is about the action of being born (by your mother), and not about the state of being alive, you need "werden" and not "sein".