Separation in direct limits of closed inclusions

The answer is no. H. Herrlich showed, in 1969, that even if you consider each $A_n$ a completely regular space, the direct limit may fail to be Hausdorff. However if all $A_n$ are T$_4$ - spaces then $X$ is a T$_4$ - space (it's not hard to prove this).

A comment about the definition of direct limit. Usually, in category theory, we call direct limit a colimit of a directed family of objects. Using this terminology it's well known that the category of Hausdorff spaces isn't closed under direct limits. You can find some examples in Dugundji's 'Topology' (a shame it's out of print). The definition you are using is very particular, so Herrlich's example is special.

In this paper D. Hajek and G. Strecker exhibit sufficient conditions for the Hausdorff property to be preserved under direct limits.