Does the Eidolon get a negative level? How does that interact with the various curing and healing rules, especially since an eidolon doesn't heal naturally?
Nothing special, really.
Negative levels work like normal, because the eidolon is a creature and nothing overrides it (specific trumps general, and in this case there is no "specific").
The eidolon does not heal naturally
is referring to the natural healing process, i.e. after 8 hours of rest characters recover 1 hp per HD (2 hp per HD if provided long-term care, see the Heal skill). Temporary ability damage returns at the rate of 1 point per night of rest (8 hours) for each affected ability score. Complete bed rest restores 2 points per day (24 hours) for each affected ability score. But we're not talking about ability damage (or drain, for that matter) here: we're talking about negative levels.
Class Level: This is the character's summoner level.
Kind of irrelevant, and on top of that taken out of context. This is used for interpreting the Eidolon table, where one column is "Class Level". Eidolons have no class levels. As with all creatures, eidolons have HD (Hit Dice). For most cases, HD and character/class level are interchangable, but it depends on context.
So what does happen for the eidolon, then?
You didn't mention the summoner's level, but because your party was fighting wights I'm going to assume that the party is around 4th level (two wights = CR 5).
Look at the table. 4th level summoner has an eidolon with 3 HD.
- If the eidolon takes one negative level, it has 3 HD (because negative levels don't actually reduce levels) and a -1 penalty on (basically) all d20 rolls.
- If the eidolon takes two negative levels, it has 3 HD and a -2 penalty on (basically) all d20 rolls.
- If the eidolon takes three negative levels, it dies.
Or, "dies", in quotation marks. It is banished to its home realm, and comes back 24 hours later. Eidolons are outsiders; in the Material Plane they exist more as a concept than as living organisms. They're weird like that.
Mind you, the eidolon will have to make a Fort save 24 hours after taking a negative level.
Energy Drain (Su):
(...) Negative levels remain until 24 hours have passed or until they are removed with a spell such as restoration. If a negative level is not removed before 24 hours have passed, the affected creature must attempt a Fortitude save (DC 10 + 1/2 the draining creature’s racial HD + the draining creature’s Cha modifier; the exact DC is given in the creature’s descriptive text). On a success, the negative level goes away with no harm to the creature. On a failure, the negative level becomes permanent. A separate saving throw is required for each negative level.