Maybe the site (GreenMedInfo.com) doesn't have their code set up right.
To get a link to show up in FB, you need to add an image source meta tag to the head of your document. The tag looks like this: < link rel="image_src" href="http://site.com/image.jpg" /> (Remove the space between < and link) Make sure you change the href to where the image is hosted, using the full URL. In order to get Share Preview and other features to read the image correctly, you need to make sure it's between the head tags in your html. It also helps to have the meta tags "title" and "description".
GreenMedInfo's site has this (I put a "*" beside the line of code that should be corrected). They aren't actually showing the full link, only part of it (missing the http://www.greenmedinfo.com/ part. :
<head>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
<title>GreenMedInfo | Alternative Medicine | PubMed | Vitamin Research | Natural Therapies | Medical Database | Disease | Toxic Chemicals | Natural Substances | Pharmacological Actions | Complementary Medicine | Herbalism | Integr</title>
<meta http-equiv="X-UA-Compatible" content="IE=EmulateIE7" />
<meta name="google-site-verification" content="Z9XuoBaKgrnyv8OA3VDNmBDsD5NQBpFKQOUH77-HGkg" />
<meta name="y_key" content="32294b99c5f0fe06">
<meta name="msvalidate.01" content="5816FF94D2DDAAF8F3A7BF020AC2DFD2" />
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
*<link rel="shortcut icon" href="/sites/default/files/wilderness_favicon_0.ico" type="image/x-icon" />
<meta name="keywords" content="alternative toxicology,consumer health advocacy,natural medicine,natural medical database,greenmedinfo,integrative medicine,alternative medicine,evidence-based medicine,complementary medicine,pubmed" />
<meta name="copyright" content="© Copyright 2008-2011 GreenMedInfo.com, Journal Articles copyright of original owners, MeSH copyright NLM." />
<meta name="description" content="Greenmedinfo.com - Natural Medical Resource - The world's only free, entirely evidence-based and solar-powered natural medical database with 17,000+ articles and growing daily" />
<meta name="abstract" content="GreenMedInfo is a free natural medicine database. It provides access to biomedical research with therapeutic values on natural substances in disease prevention and treatment" />
<meta name="revisit-after" content="1 day" />
<link type="text/css" rel="stylesheet" media="all" href="/sites/default/files/ctools/css/dee519136596ee9b0000de9fd4f16965_0.css?7" />
<link type="text/css" rel="stylesheet" media="all" href="/sites/default/files/css/css_016c4901af5a1deb8cb5a186add650be.css" />
<script type="text/javascript" src="/sites/default/files/js/js_6ec6b8f834a4bb18214979572dbee927.jsmin.js"></script>
<script type="text/javascript">