Answer to "Will you be able to provide proof of your identity and employment eligibility if you are hired"?
Unless you're already eligible to work in the US (e.g. as a permanent resident), you'll have to answer the second question with a "no."  

The three aspects of US employment/immigration law that you need to know about are that 


Newly hired employees are required to prove that they're eligible to work in the US by filling out an I-9 form and providing the required documentation.  Most universities  than use an online service called e-verify to verify the information on the I-9 form.  You'll typically be required to provide this documentation immediately upon being hired (not six months later when your first semester starts.)  You're being asked if you can fill out the I-9 form.  
Practically all of the options for a visa that would enable you to work in the US require sponsorship by the employer.  This sponsorship requires the employer to declare that there were no qualified applicants that weren't already eligible to work in the US and obtain an authorization from the US Department of Labor.  The process takes weeks to months, and can cost thousands of dollars.  If you say that you're good to go, and the employer later finds out that they'd have to sponsor you for a visa, they may simply withdraw your job offer rather than sponsor you for a visa.   
It's not legal to discriminate between various classes of people who do already have the right to work in the US (e.g. between permanent residents vs. citizens.)  The particular wording of that question is designed to protect against that kind of discirimination.  


Going beyond the "yes/no" answer to this question, you could provide additional information that could be helpful.  For example, if you hold Canadian citizenship than its easy for you to be employed on a NAFTA visa, and you wouldn't need to be sponsored for an H-1B visa.  On the other hand, if you're currently on an expiring J-1 visa with a "return home" requirement, it could be difficult for them to sponsor you for an H-1B visa.