I am not a USA resident but I've worked the last summer in the USA so now I want to get my state income tax back. I know that the money is coming in form of a check on my name. I need somebody else to cash it for me while still in USA without me being actually there. Is there any formal power of attorney/proxy document that I can make in my country to give it to my friend ?
It would be better to use a bank account and have the refund deposited directly to it. But you said you never had a bank account, so that may be a problem.
Another option is to have the refund check mailed to you, and you deposit it in your local bank, converting to your home currency (or not, depending on local laws).
Generally, for another person to cash a check made out to you - you need to endorse it first. Physically, on the back of the check. That means you have to see the check.
Specifically with tax refund checks there's much more scrutiny since there's a lot of fraud going on with regards to tax refunds. Thus, I doubt a bank would allow a third party cash a check made out to you, without you actually being present there.