Everyone has their own concerns, don't they? or,
 Everyone has their own concerns, doesn't he? or,
 Everyone has their own concern, don't they?
Or else? Please advise.
In , 'their' and 'they' are politically correct, but grammatically suspect.
In , 'he' is politically incorrect, but grammatically correct.
The rule of thumb is to try to use language that isn't suspect. For example,
They all, every one of them, have concerns, don't they?
In ,  and , "their" represents a singular pronoun (i.e., combined she and he). Its referent is singular 'Everyone'. Again, it's politically correct, but grammatically suspect. That is, Standard English does not have--or at least not yet--a singular pronoun "their", but speaker do in fact use "their".
In , either 'concern' or 'concerns' is fine:
Every single one of the girls has her own concern.
=> Each girl has one concern.
Every single one of the girls has her own concerns.
=> Each girl has more than one concern.