I am a student on an F1 visa. I pass the substantial presence test, and I have spent more than 5 calendar years cumulatively on an F1 visa. Should I file just the 1040 on Turbotax? Do I need to submit anything else to prove my eligibility?

If 2016 is your sixth year staying in US, and you stayed at least 183 days during 2016, you are considered as a US resident for tax purposes and required to file a Form 1040, like everybody else.  

As TurboTax is designed to support most of the tax forms for US citizens and residents, at filing, you do not need to submit anything to prove your eligibility.