I would suggest you rewrite either of the following: "Free from artificial colors and flavors." "No artificial colors or flavours." I think these sound more natural.

You're speaking in the general sense here, so use colouring/flavouring. By adding "-s", you're suggesting that you have a list of some sort in mind - specifically "different typeS of COLOURING". Since we (the readers) have no idea what's on your list, then you should use the general sense: colouring, flavouring. (Pardon me; I'm using the UK/Aust spellings for these words. Your US spellings are still correct, just keep the style consistent.)

"free from artificial coloring or flavoring" is perfectly correct. "Colorings" is wrong!

we can use plural to emphasise that it's free from any sorts of colouring, BTW singular form can also be used.

You could use plural (colorings,flavorings) or singular (coloring,flavoring) forms. What may need to be change is "or" to "and" but that depends on the context. Customer: Are there any artificial colorings or flavorings in this soda? Salesman: Our products contain no artificial coloring and flavoring!