Carbonyl carbon electropositivity?
Is the carbon in a carbonyl functional group electropositive?


Electronegativity and electropositivity are usually used in a relative sense, X is more electronegative than Y.  The carbonyl carbon in acetone is more electropositive than the central carbon atom in isobutane, and less electropositive than the methyl carbocation carbon.


  because he defined electronegativity earlier as the ability to
  stabilize negative charge


Weak definition, wouldn't you say; that's why you won that argument.  Atoms can be electronegative in the abstract, charge doesn't need to be involved.  The statement, "fluorine is more electronegative than carbon according to the Pauling Electronegativty scale" is accurate and understandable, yet no one is describing a situation where charge stabilization is involved.


  because electrons cannot stabilize other electrons.


I understand what you mean, but you should know by now that you can't make such sweeping statements.  Don't the electrons in one of the double bonds in 1,3-butadiene interact with and stabilize the electrons in the other double bond?


  Can we describe the electron deficient carbon as electronegative


Again, I understand what you mean, but... Yes, it is more electronegative than the carbon in an alkane, but not as electronegative as $\ce{F+}$.


  Is electronegativity the ability to stabilize negative charge or the
  ability to attract electrons within a covalent framework? I guess the
  two go hand in hand - i.e. you wouldn't attract electrons if you
  weren't able to stabilize them


Yes, the two go hand in hand.  More electronegative atoms are better able to stabilize negative charge than less electronegative atoms.  It's all about lowering the overall system energy. 


  As a side question: can we calculate EN for ions?


I'm sure there is a way.  For example, using studying the $\ce{^{13}C}$-nmr spectrum of various carbocations leads to a relationship of (somewhere around) 180 ppm shift/ electron.  So you can estimate the fractional positive charge on any carbon atom, the carbonyl carbon in acetone for example.  Once you've estimated the amount of charge on an atom, I suspect there is a way to relate it to EN.