Is the health premium tax credit (1095-A) only for itemizing, or is it in addition to the standard deduction? I did not receive advanced payments.

No. The Health Premium Tax Credit in not only for itemizing. It is included with standard deduction. Would you clarify, did you received a 1095-A?

Yes, I received a 1095-A [This is the first year I did not get health insurance from my work, I purchased from the marketplace and pay a monthly premium out of pocket. I did no get an advance on the credit.] 

Is the health premium tax credit only available if you are itemizing, or is it also available if you take the standard deduction? 

TurboTax is claiming I need to pay for an upgrade in order to file my taxes with the 1095-A. However, I think the standard deduction is better for me if it's a choice between the standard deduction without the premium tax credit or itemized with the premium tax credit. I don't want to pay $60 for the software to tell me to file a simple tax return with just the standard deduction. 

You can get the credit if you are taking the standard deduction - it is not an itemized deduction.   The premium tax credit requires Schedule 3 and Form 8962 which require Deluxe anyway.   So if you have more itemized deductions, you may as well itemize.