If I took the train to Verdun and rented a car, would it be reasonable to try to do both Verdun and Meuse-Argonne in the same day, or would I be better off to spend the night and break it into two days?
The American Meuse-Argonne Cemetary is 26 miles northwest of Verdun, so I'd think that would be doable.
http://www.abmc.gov/cemeteries/cemeteries/ma.php
Depends on what you are planning on doing in each area. Verdun can be done in 1/2 day, or you can spend a full day. If you're re-tracing someone's steps, or want to spend hours looking through the stereoscopic photos, then you might be better off breaking it into 2 days. If you just want to hit the highlights, one should be enough - just make sure you get the earliest start possible. (And I'm assuming you aren't going in the dead of winter when there's little daylight. If you are - you'll need two days just because of the limited daylight hours.)