Can you claim active military status for 2020 if first day of duty was 12/28/2020?

Yes, as long as your were Active Duty Military on December 31st, you can claim active duty military status in TurboTax.

Indicate this in the 'My Info' section of your return.

Click this link for more info on Military Tax Tips.

...hmmmm...might not work.

Unless you get a 2020 military W-2 from DFAS, the software may not accept you for getting free software for Deluxe, Premier, or SE versions.  Of course, if you can use the "Free" version anyhow, it make no difference.

first day of duty was 12/28/2020

The final military pay for tax year 2020 was posted on Wed, Dec 30 for most, Dec 31 for others. If 12/28/2020 was your first day of basic training, then I doubt you got a DFAS issued W-2 for 2020. But it's possible you did if you were paid some cash up front by the military, which is what they usually do your first "official" day of training.  The day you arrive at basic training is not your first official day.