We have begun playing the guild expedition in a guild with only 2 members. We have progressed to the 2nd level, paid to go on, and have begun to proceed there. Seems to be working fine so far. I assume it will also work for a single member guild. It seems the only criteria is for there to be a guild.
The nice thing about it is that each of our stats show up in our respective PvP areas and accumulate there as well as building the guild's standing. I don't need to attack and plunder my neighbors to do the PvP, nor need do the GvG either. I can work for the guild and help myself all at one time without getting too involved.
If they created their guild after the week's GE started they might of thought that being a 1 member guild meant they could not participate. Otherwise I would think (as some others suggested) that it was goods related (they may not have a very good trade network so find it harder to get the goods they don't make) or that they realized to gain the most benefit from GE would require a larger guild since you'll advance through the levels faster with more people doing GE.