Can I say that $E[X | c] = E[c | X]$, when $c$ is a constant and $X$ is a random variable? $X$ is a random variable defined in a sample space
$c$ is a constant $\in R$ (real numbers).

Maybe translating to an applied example will help illustrate why the equality typically doesn't hold (see the other answers for the special cases in which it does). I will use conditioning a bit hand-waivingly here as an information set rather than as proper conditional distributions.
Take $c$ to be the constant $\pi$ and $X$ the outcome of a coin flip, heads or tails. If you knew that $c$ equals $3.14...$, would that alter your expectation $E(X)$? No, I suppose, so $$E(X)=E(X|c)=1/2$$ (assuming a fair coin).
Conversely, if you knew the outcome of the coin flip, would that alter your expectation regarding $E(c)$? No, I suppose, so $E(c)=E(c|X)=3.14...$
So my hunch is your professor wanted to illustrate that conditional expectations do not "commute".
