Generally, as a non-resident, you would need to file a CA return if:

• you were employed in Canada;
• you carried on business in Canada; or
• you disposed of property in Canada.

None of these seems to apply to your situation. HOWEVER, you may still want to file a return if you want to claim a refund. 

TurboTax Standard is our most popular version, which should take care of your situation. However, non-resident filings are on the complicated side, so you may want to get one of our experts to review your return.

Lastly, if you decide to file a return, then yes, you will need an Individual Tax Number (in lieu of the Social Insurance Number). You will need to contact CRA for this.

I was only paid once, and I thought it was the US company that paid me (my pay is direct deposited). I think the Canadian company listed on the T4A-NR must be some sort of a subsidiary of the American company. My contract is to perform services for the US company. It looks like I had 15% of the gross income withheld.

I think my original answer still applies in your situation. If you think you are going to get a refund (because they've over deducted on your Canadian wages), then you should file a Canadian return. Otherwise, I think you just need to report the amount on a US return and you may be entitled to a foreign tax credit. The T4A-NR is proof of taxes withheld in Canada for the purposes of the foreign tax credit.