Translation:Everyone usually has flu in winter.
Yes, it can, but need not, everyone gets flu in winter is the normal way to say it.
for those who might be interested, I just like to bring a nuance with the use of "thường xuyên" vs "thường". in the initial sentence "mọi người thường xuyên bị cảm cúm vào mùa đông", the speaker implies that a lot of people are sick during winter. if we change "thường xuyên" for a simple "thường", the speaker will then mean people are more often sick in winter in comparison to other seasons.