Everyone would take some misunderstanding, take it or leave it...I would be careful what to say about, "Our God," or "Everyone's God." Some may find it insulting as if you were referring it is your God only, not theirs, or our same God, or too many Gods, or animal's God, so on... Some may think you were referring our God that there is only one God in any kind of reglious or none at all that includes non-reglious and non-believers... Everyone, even non-believers, non-reglious do have a God, with or without thier knowledge...Remember God works in mysterious ways...think about it...and for that Holy Bible, Ten Commendments, etc...not everyone knew if it does exists or not. For a perfect example...Our Father art in Heaven...Our Heavenly Father, Our Lord, In our Father's name...There are quotes in the Bible, any other Bibles, or on your own, those quotes doesn't always meant for Our God or Their God only...
I respectfully disagree with you about how people will find it insulting if I say "Our God" It's the same as "Our Father" "Our Lord" "Our Heavenly Father" "Our Heavenly God" I'm not here to pleased anyone, If I want to say "Our God" I have that right.
Of course non-believers, and non-religious people do have a God, but they chosen not to believe in one God, some believe in gods, some believe in idols, some believe in none.