Most publishers and journals allow the submission of manuscripts that have previously been posted on a preprint server (notable exceptions are NEJM and JAMA, see Wikipedia entry "List of Journals by Preprint Policy"). Thus, it is usually not a problem to submit a paper rejected for JMIR to another journal (although we would certainly prefer if the paper would stay within the JMIR family, see How do I request a manuscript transfer to another journal?).
Our preprint server will also automatically detect when a manuscript is published elsewhere and add a link to the Version of Record - be it in a JMIR journal or any other journal - on the preprint landing page, linking to the final published version.
When authors publish on our preprint server, authors retain the copyright and also do not publish under a Creative Commons license (which would prevent later publication in a non-cc journal, i.e. authors can still assign the copyright to another publisher).
Our preprints are also - on purpose - not indexed by Google.
If plagiarism software of the other journal picks up the JMIR Preprint (although the settings should usually prevent search engines from indexing the preprint), please check their preprint policy and point out that prior "publication" on a preprint server is allowed as per their own policy.
If you still have problems, file a support ticket with firstname.lastname@example.org, stating the journal name and quoting what the other journal/publisher says. We will either negotiate with the other publisher/journal, or remove the preprint from our site. However, a preprint is supposed to be stable and citable (we also assign a DOI), so this would be only the last resort. See also How do I change my preprint settings or availability of preprints?