Can the enthalpy change of a reaction be changed, and if so, will this affect the value of the equilibrium constant?
Short answer: No and yes. 

Enthalpy is a state function. That means that it depends only in the current "state" - the identity of the substances, their concentrations, pressures, etc. That means that the absolute enthalpy of some substance is independent of how you produced it. However, that substance has different enthalpy in different states: gas versus liquid, pure versus in solution. If you change the state of the product(s), you with change their enthalpy. Same goes with any of the other thermodynamic potentials: entropy, internal energy, free energy. 

And that of course will alter the equilibrium situation.