Does increasing electronegativity imply more covalent character?
The main force responsible for holding together covalent bonds  and ionic bonds is different. Ionic bonds are held together through electrostatics. A transfer of electrons causes one atom to be positively charged and another one to be negatively charged. In other words, electrons are given away or taken away by atoms. Covalent bonds are held together due to the delocalization of electrons, i.e. electrons are shared by the atoms. For more detail look at What makes the difference between ionic and covalent bonds? (physics.se)

There is no meaning to the statement that an element is covalent. The reason is that the nature of a bond (metallic, covalent, ionic) is determined by the elements participating in the bond. Sodium may form a metallic bond with itself, but it will form an ionic bond with halogens. Halogens will form covalent bonds with themselves but will form ionic bonds with alkali metals. Furthermore, the conditions can determine the nature of a bond. For example, hydrogen is metallic under high temperature and pressure. 
Just a side note, compounds can be ionic. I recommend that you read this thread explaining the difference between compound, element, mixture, etc: What is the definition of of &#39;compound&#39;, &#39;mixture&#39;, &#39;element&#39; and &#39;molecule&#39;?

Tldr: Wikipedia is not being exact. Saying that an element is covalent is inaccurate because the terms covalent and ionic refer to bonds not elements.