Why use EBIT instead of Net Income or EBT for FCF calculation?

Before we get into the equation, we should first clear up what we mean by FCF.

There are two types of free cash flow that are looked at: Free Cash Flow to the Firm  (FCFF) and Free Cash Flow to Equity (FCFE). The difference between the two is that FCFF represents the amount of cash that a company generates which it can use to pay either its creditors (debt holders) or its owners (equity), while the FCFE represents the amount of cash that the company generates after servicing its mandatory debt for its owners.

The next question that you might have is why would we care about unlevered free cash flow. Simply put, before putting debt on a business, you need to evaluate the business's ability to service the debt and unlevered free cash flow is a good metric (although not the only one) to look at when making such decisions. Debt, also known as leverage or gearing, has a number of benefits to equity holders given it lowers taxes payable (because interest is tax-deductible) and increases IRRs by reducing upfront equity checks. Finally, if we take our FCFF and discount it at the company's weighted average cost of capital (WACC), the result will be the company's implied enterprise value. Remember that enterprise value is the value of a company to all of its claimants (both debt and equity). As such, when solving for enterprise value, we need to look at FCFF to keep the comparison apples-to-apples.    

Now to your main question, it is important to clarify that the equation of FCF = EBIT (1- Tax Rate) + D&A - Increase in Net Working Capital - Capex represents free cash flow to the firm (because it is still before interest expense). You could just as easily use the equation FCFF = EBITDA - Cash Taxes - Increase in NWC - Capex or host of other derivations of the formula to get to the same result. The important thing is that you think through the cash flow statement (building a cash flow waterfall can be helpful in these scenarios). In the cash flow from operations section of a cash flow statement, D&A and Increase in Net Working Capital are usually adjusted out, while capex is represented in the cash flow from investments section. Again, we are just trying to find whatever cash the company generates that it can use freely. Some thought will also be needed on a company-specific level (a question left to the reader:  is it fair to consider all capex to be a mandatory use of cash? How should we consider expansion capex when thinking about free cash flow?) to really get to the bottom of the question, what is the FREE cash flow?

Finally, to your question why we don't use net income instead? Honestly, sometimes we do if we need to get to free cash flow to equity (because net income has already been impacted by interest). The most common place you'll see this is in LBO modelling tests, where you only cares about calculating returns to the sponsor. 

Hopefully this cleared up the concept - all the best with your summer recruiting!