I do hope that was said tongue-in-cheek as I know quite a few vegetarian Orthodox (some of who gave up meat for health reasons and one or two that just can't stand the taste or smell of meat).
Seriously, though, the point to the fast-free weeks is not just to relax the fast for those who want the break, but also to have times where fasting simply isn't allowed. The Synaxarion entry from the Lenten Triodion for the week of the Publican and the Pharisee says the following: "So that we can learn to avoid the pride of the Pharisee by following our own self-imposed and self-directed fasting practices -- instead of the moderate and time-tested fasting traditions of the Church -- the following week is fast-free." The week was specifically picked as one of the fast-free weeks because the Pharisee bragged that he "fasted two times a week."
My common sense interpretation of all of this would be that for anyone not eating meat for health reasons (although I have a hard time thinking of any health conditions in which the smallest portion of meat couldn't cross one's lips without negative health consequences) this wouldn't apply. Also, if someone just doesn't like meat, that wouldn't fall under fasting.
If a layman is avoiding meat as an act of asceticism, then the directive to stop fasting during these weeks would seem pretty clear, since that layman is by self-direction choosing to follow the monastic practice of never eating meat. But if it has nothing to do with fasting, then one can stop fasting in other ways -- eat fish and dairy products on Wednesdays and Fridays on those weeks, for instance.
On the other hand, if someone thinks that being vegetarian brings one to some sort of a higher plane, and looks down on those who aren't vegetarian, then again my common sense interpretation of the Church's directives would be that it would do that individual good to have a hamburger or steak during the fast-free weeks. Ultimately, it's mostly about our attitudes -- we have a tendency either to be gluttonous on the one hand, or to be prideful about our self-control on the other...
I was responding to someone who said that she was a vegetarian but missed eating lamb!