By all means paint a plywood coop floor (even if you're going to put down sheet vinyl, IMO - *some* moisture will still get under there, I guarantee). It will last longer vs rot when you have unsuspected damp spots under the bedding, and in places where the chickens kick away bedding to expose floor, it is ever so much easier to get trompled poo off *painted* wood than naked wood.
Make sure to prime first, and then apply several THIN coats of (preferably exterior semigloss) paint, not fewer thicker coats. Thicker coats start peeling pretty soon; multiple thin coats last.
Good luck, have fun,
Pat