First of all, what is all this talk about sodium "wanting," or "prefering" to do anything? Atoms don't have feelings.
As far as gaining or losing electrons, if a sodium atom loses an electron, and a chlorine molecule happens to pick up that electron, NaCl(s) can form. This is way lower in potential energy than the reactants, so this reaction is spontaneous. Find lattice energy in the index and look at the diagram for this in your text. The student is correct in that E must be absorbed by Na to form Na+, and a little E is released in forming Cl-, but the majority of the E released is from the two ions coming together.
As far as IE goes, if you check your gen chem text, the ionization energy for every single element is +, meaning it requires E to separate atoms from electrons. Does this mean that cations don't exist? Of course not.
EA and IE are useful concepts/data to help explain what is going on in a chemical reaction, but these numbers are for gas phase reactions of atoms.
About NaH: there isn't much support for this compound to be labeled as Na- and H+. This compound reacts with water (and other proton sources) to form H2 gas, leaving behind an anion (OH- if it's reacting with water).
There is the odd Na- H+ compound, but someone worked really hard to make this: http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&list_uids=12022811&dopt=Abstract
So the original poster did a nice job questioning her prof, and trying to reason stuff out. But please, no more atomic feelings! Atoms gain or lose electrons to form compounds, but they don't do this unless enough energy is returned by new bonds forming. Check out the difference in the IE's of Na and then of Na+ and you can see the difference.
Yes good point NaH is sodium which has gained an electron, -1 oxidation state, 1s2.
So what is the right answer to the original question? I would say that sodium tends to lose an electron, what do others think? Should the answer be both lose and gain an electron? (maybe depends on the level of the student?)