If I queue as an Arms warrior, will I be eligible for both the 1h Axe and 2h Sword that Elegon can drop? I believe as Fury I can but I currently lack a second 1h or 2h weapon, and am not familiar with the spec.
As Arms, no. Arms doesn't use one-handed weapons, so the one-handed weapons wouldn't be flagged for Arms.
If you're willing to use Elder Charms on LFR, though, you can switch specs right after the kill and when you use the charm it will roll for the spec you're in at the time (not what you were in for the kill).