Do I have to file taxes in both US and CA if I received a T4A-NR for working 14 days in BC? (I'm a US resident/citizen). How do I avoid paying taxes twice?

To avoid double taxation, you will need to report the amount you earned on your US tax return as foreign income subject to the foreign tax credit.

I am not familiar with Canadian taxes and do not know if the Candian tax withholding on your T4A-NR is considered a final tax or if you need to file a Canadian return. You may wish to check TurboTax Canada to see if they can assist or call the Candian Revenue Agency and see if an agent can advise you of any Canadian tax filing requirement.

As you are a US citizen or resident, you will have to file a US tax return reporting your worldwide income no matter where it is earned or received.