You can get at this information, but it's messy.

Using the `/posts/{id}/revisions` route you can query for the list of all [revision](https://api.stackexchange.com/docs/types/revision) events for a particular question, which includes close/reopen events. These have `revision_type = "vote_based"` which you can filter the results by, but so do delete/undelete and protect/unprotect events, meaning you ultimately have to parse the `comment` field for `<b>Post Closed`, etc.

You can see all of the various different states in action in the [revisions of Jon Skeet Facts](https://api.stackexchange.com/docs/revisions-by-ids#pagesize=100&ids=9134&filter=default&site=meta&run=true), which corresponds to the [on-site question history](http://meta.stackexchange.com/posts/9134/revisions).