Assuming that the diagonal map $X\rightarrow X\times X$ is a cofibration. Is it true that the diagonal map $\Sigma X\rightarrow \Sigma X\times \Sigma X$ is a cofibration? (Where $\Sigma X$ is the reduced suspension of $X$.)
Yes. See Gaunce Lewis' paper
By definition $X$ is locally equiconnected (LEC) iff the diagonal map $X \to X \times X$ is a cofibration (of unbased spaces). It follows that $X \times I$ is LEC, which implies that $\Sigma X$ is LEC, by the Dyer-Eilenberg adjunction theorem for LEC spaces. See Theorem 2.3 in Lewis' paper.