You don't explicitly say (this being an internationally visited and populated site), but based on your question, I will assume that you are in the US.
For the question you asked: Is the company the government? If not, then NO, you cannot successfully sue a company (or person for that matter) for violating the freedom of speech granted by the First Amendment to the US Constitution in any circumstances whatsoever. (Sorry, this is a pet peeve of mine).
The US Constitution does not bind or restrict any private* individual or company, in any way. (Here "private" means "non-governmental; a "public(ly traded) company" is still considered a "private" entity in this context). The US Constitution exclusively deals with four things: How the US Federal Government operates, powers of the government, and restrictions of the government, and the definition of treason (which arguably is itself a restriction on the power of the government, by denying them the ability to define treason themselves).
The First Amendment itself is explicit about this restriction:
Congress shall make no law ... abridging the freedom of speech (emphasis mine).
Note that, while the First Amendment does not mention acts of the President, this is because the President's Constitutional powers are quite weak and limited; What powers the President does have and usually uses are granted to the office by laws passed by Congress, and so the restriction comes with them, as Congress cannot delegate to the President powers that Congress themselves do not possess).
As such, no company can be sued for violating the First Amendment (or any portion of the Constitution, really) because it does not apply to them.
Now, there may be laws passed by relevant legislatures, but these are dependent on your jurisdiction (e.g. state). However, as a general rule of thumb this would be legal. Turning down a candidate based on what they say in an interview is the point of having an interview; Turning down an candidate for saying something in an interview that could potentially leave the company liable for a lawsuit under the theory of vicarious liability is only good common sense.