I am a non-resident. After checking the docs sent it did not include the 1040NR-EZ form for non-resident but 1040. Would TurboTAx support this kind of amend?

No, TurboTax does support the 1040NR.  As a nonresident, you will need to visit our partner site, sprintax.com. They specialize in Non-resident returns and will be able to guide you through what all you need to do. 

Thank you for your answer. However, I have paid TurboTax for their services, and responded to the questions honestly. And now I am supposed to pay another services because Turbotax was not able to filter my case out. How can I get reimbursed in these circumstances?

You will need to call into Customer Support.  Intuit Phone Support: 1-800-446-8848

If you are not in a big hurry, you may want to wait until Tuesday to call since the hold times are currently long and will be through tomorrow night.