Relationship between enthalpy change and endo/exothermic reactions
Yes the region x represents Enthalpy change.The relation between Enthalpy change and endothermic reaction  is that if a reaction is endothermic then Enthalpy change is +ve and if the reaction is exothermic then the Enthalpy change is -ve.

In an endothermic reaction the amount of energy you supply is lesser that the amount of energy released when product is formed so the amount of energy you supply to transform the reactants to activated complex will be more than the energy released when the activated complex transforms to product.

In the graph you have provided the reaction in forward direction is exothermic and in backward direction is endothermic. Whichever direction you take  activation energy will always be more than the Enthalpy change because Enthalpy change is the difference in the Enthalpy of the products and the reactants whereas the activation energy is the difference in the Enthalpy of the activated complex and the reactants. The Enthalpy of the activated complex is always more than the Enthalpy of the products and the reactants so both these quantities cannot be equal.

For both exothermic as well as endothermic reactions it not a necessary condition for the activation energy to be equal to the Enthalpy change as activation energy is always greater than the Enthalpy change.