Both are accepted, but after a search on Google you get: 'opinion about': 1,220,000,000 results 'opinion on': 10,160,000,000 results So according to Google 'opinion on' is used about ten times more...
It would be more correct to say "Everybody has his/her opinion."
The use of their in this sentence would be correct, though it is widespread in American speech. Originally, one would say "Everybody...
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