Looking at the comment thread above, I believe a bit more elaboration can help, so here it is
Why is electron gain generally exothermic?
Think of a unit positive sphere and a unit negative sphere. When is their system most stable? It is when the spheres are closest to each other. Why is that position stable? Because if we slightly disturb those spheres to take them farther from each other, they'll again come back to their original position, due to their attractive forces.
This stable position is an energy minima for this sphere system. When a system goes from a higher energy state to a lower energy state, it loses energy.
The same is the case with the system of the negative electron and the positive nucleus pair. The energy of an electron in vacuum (isolation from the nucleus) is zero (because it cannot feel the nucleus' attractive pull). But when it occupies a definite energy level in the atom, the energy of the system reduces. The extra energy is then released, which is why you find that the electron gain is generally exothermic.
Why is energy required to remove electrons of elements like $\ce{Be, Li}$ that naturally want to lose their electrons?
I assume you mean to say that $\ce{Na}$ easily loses electrons in presence of, say $\ce{Br}$ (assume $\ce{NaBr}$ ionic), even though $\ce{Na}$'s ionization energy is positive. That is because the formation of $\ce{NaBr}$, or any ionic compound in general, involves a number of steps, each involving a definite energy change. Ionization of the sodium atom is just one of those several steps!
This is called the Born-Haber cycle. The general idea is that you start with solid sodium and bromine liquid, and eventually form solid sodium bromide. The net enthalpy change for this entire process is negative, or exothermic, hence it is favorable. The positive ionization enthalpy is compensated by the other processes which release energy.
However, in vacuum, the only process that occurs with the isolated (other ions are absent) gaseous sodium atom is to remove an electron. That process - which defines ionization energy - is still endothermic, because you need to oppose the nucleus' attractive pull.
Elements with fully filled and half filled highest occupied orbitals have positive electron affinity (endothermic). Why is this true?
That is true because elements with fully filled and half filled highest occupied orbitals are stable (you can read why that's true here). Adding an electron to this atom would make it unstable, thereby increasing the atom's energy. Now, don't get me wrong, when you add an electron to the atom: the energy of the nucleus-electron pair still decreases, but the rise in energy due to the instability is much more, causing a net increase in the energy of the atom.
Hence, in such cases, the electron gain process is endothermic.
