This is a complicated question and therefore difficult to give a simple answer to. If you had followed my past arguments or attended my talk about electronegativity at the Philadelphia ACS meeting, then you might know Pauling was trying to explain atomic reactivity. Simply, fluorine is the most reactive element (as an electron acceptor), hence most electronegative. Oxygen is the second most reactive, not chlorine, bromine, or iodine. Carbon is an intermediate atom between an electron donor and acceptor and is very effective being able to share electrons.
The reactivity of carbon or butane should be placed in context of reactivity. Will or can it react with another element and release energy as a result? Ethylene or acetylene can react with hydrogen to give ethane plus heat. This teaches us about the relative energy levels of those forms of carbon and hydrogen. Butane does not react with hydrogen to give a new compound and release heat. Butane will react with oxygen or fluorine to give new compounds plus heat. Thus even though carbon-carbon bonds are stable to hydrogen or carbon, they are less stable than C-F, C-O, H-F, or H-O bonds.
This is not necessarily a reflection of the stability of carbon-carbon or carbon-hydrogen bonds per se, but it is also a reflection of the reactivity of oxygen and flourine. Perhaps a measure of bond stability might be compared with the energy required to form a new bond. It is relatively easy to form new carbon-carbon bonds, but relatively difficult to reform flourine, hydrogen, or oxygen from other combinations. if thought of in this way, we could recognize that carbon is good at sharing electrons in chemical bonds, but also capable of forming new bonds with very reactive elements.
In my talk, I showed two charts of the same reaction. Typically, the convention is to calculate the energy differences from the elements (set at zero). This makes HF seem very stable. If plotted with the products as arriving at the same energy levels, then flourine appears to start with the greatest energy. This is the core of electronegativity. If the posters question is considered from this perspective, then it is not a question of electronegativity that determines the stability of carbon bonds, rather the inate properties of carbon. Carbon is stable because it is stable. It can only react with something more reactive than itself.