I've got an unusual situation (at least as far as I've been able to search). I received a 1099-B from Computershare. There are several transactions similar to this scenario: I ordered a sale of 150 shares. One-third are marked Short Term and include a cost basis. One-third are marked Long Term and include a cost basis. The final third shows nothing on a cost basis and is checked as non-covered.
I've been looking at purchases/reports in Quicken to find that final cost basis, I've reviewed the sales order details (which shows a cost-basis on everything, with no semblance of order to try to reconcile to one of these thirds). I'm wondering if it's because they could be matched shares from my employer, but even then there were purchased at the same time I purchased my shares, so there's a cost basis. Applicable check box on form 8949 shows X on these lines. I'm just at a loss what to do now.
It sound like you're on the right track, that those shares with no cost basis were possibly 'sold to cover' by your employer to pay tax on the other shares you sold. Do you have a 'Supplemental Statement' from the broker that shows a cost basis for these stocks?
You could reach out to your employer to confirm this.
Here's some detailed info on Employee Stock Purchase Plans and a helpful video on How to Report Sale of ESPO and RSU stock.