The term 'noncovered' simply means the broker does not know your cost basis and therefore is not reporting that to the IRS. Your cost basis is the market value of the shares on the day it vested (this is included in your W-2 income and therefore becomes your cost basis per share).
The stock, once you have complete rights to the stock is short term if held less than one year. The stocks sold at the vesting date to cover your taxes will be short term holding period.
Holding for more than one year is long term and receives special capital gain tax treatment.
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