The "nextSibling" can be a tricky thing. So try to make it simpler by replacing it with a direct reference:
1. replace [this.nextSibling] with [myvideoid]
2. insert [ id="myvideoid"] into the second <div> tag as below
If you need to use the photobucket's video you should replace all the code within the <object> ... </object> with the code which is suitable for the photobucket (you can find it at the photobucket's page as "HTML code" in the "Share this video" box). The bold part is what should be added to the code line from the photobucket. Though I'm not sure if my replacement for the <object> code would work correctly.
Code:
<div onclick="myvideoid.style.display='block'; this.style.display='none'">
<img src="image.png" style="cursor:pointer" />
</div>
<div id="myvideoid" style="display:none">
<!-- Embed code here -->
<object width="600" height="361">
<embed width="600" height="361"
type="application/x-shockwave-flash" allowFullscreen="true" allowNetworking="all" wmode="transparent"
src="http://static.photobucket.com/player.swf"
flashvars="file=http://vid444.photobucket.com/albums/qq167/rockzmom/MasterRussian/364b632a.mp4">
</embed>
</object>
<!-- End Embed code -->
</div>