I have seen this happen too, and the easiest way to fix it in an existing document, is to modify the Word Style “Endnote Bibliography” and make sure this is the style assigned to the whole of the bibliiography and to a trailing extra paragraph.
To do this (at least in Word 2010 - I haven’t had much experience with this in 2013) – Open the list of styles (in 2010, this is the little corner below the styles horizontal list on the home ribbon), it will open as a list. Now select one of the references and see which of the styles it has adopted, -If it isn’t “Endnote Bibliography”, then you should apply that style. Now edit the line to appear the way you want it to appear using format paragraph settings and remove any page break assignments. right click in the paragraph and choose “styles” and Update “endnote bibliography” to match selection. Select all of the paragraphs in the bibliography, and apply this style to them. Also make sure there is a paragraph between the header paragraph and the bibliography and after the bibliography that is also assigned to this word style.
In EndnoteX7, new “Endnote Bibliography” and “Endnote Bibliography title” style are added to the list of styles in Word as soon as a bibliography is generated. It might be easier to delete the bibliography, add two paragraphs after the header, and make sure the paragraphs are either normal, or predefined as “Endnote Bibliography”
To avoid it happening I always have a Body text/normal paragraph following my manually inserted Bibliography heading. And this random “pickup” of the wrong paragraph formating doesn’t appear to happen. – I find it happens in documents that already have a number of styles defined – like in a thesis, than in a clean document. Particularly those that were originally formated with X6 and then with X7. I have not been able to put my finger on the exact conditions, but it happened to at least one of my students with Endnote X7 (where the new undocumented feature is to utilize these “word style templates” was introduced - this isn’t available in older versions of Endnote).