It seems to me like the proof for Abelian categories works here too. Consider the following diagram where $d = (id,-id)$, the middle vertical map is the fold (or $+$) map, and the square on the right is a homotopy pushout:
$$ \begin{array}{ccccc} X & \xrightarrow{d} & X \oplus X & \xrightarrow{(f,g)} & Y \\ \downarrow & & \downarrow & & \downarrow \\ 0 & \to & X & \to & C \\ \end{array} $$
From the second square being a pushout, $C$ is the coequalizer of $f$ and $g$. If we check the first square is a pushout too (I'm blanking on how to do that at the moment), we get that the outer rectangle is a pushout and so $C$ is also $\mathrm{cofib}(f-g)$.
Here's Akhil's proof that the left hand square is a pushout:
$$ \begin{array}{ccccc} X & \xrightarrow{\pmatrix{0&1}} & X \oplus X & \xrightarrow{\pmatrix{0 & 1 \\ 1 & -1}} & X \oplus X \\ \downarrow & & \downarrow \rlap{\scriptstyle\pmatrix{1\\ 0}}& & \downarrow \rlap{\scriptstyle\pmatrix{1\\1}}\\ 0 & \to & X & \xrightarrow{1} & X \\ \end{array} $$
The outer rectangle is what we'd like to show is a pushout. The right hand square is a pushout because both horizontal arrows are isomorphisms, and the left hand square is a pushout because it is the (pointwise) direct sum of these two (where all the maps $X \to X$ are the identity):
$$ \begin{array}{cccccc} 0 & \to & X & & X & \to & X \\ \downarrow & & \downarrow & & \downarrow & & \downarrow \\ 0 & \to & X & & 0 & \to & 0 \\ \end{array} $$