The Mad Greek is a chain that, per google, was started in Southern California. I have stopped at the original (1974) Mad Greek in Baker, CA, many, many times; it is halfway between LA and Las Vegas, and always busy.
If the Moscow diner is at all similar, the menu is pretty extensive. It almost reminds me of a Jewish deli-type restaurant, with the emphasis on traditional Greek--rather than Jewish--dishes. But almost anyone can find something to eat there: not specifically vegetarian, but I assume a vegetarian could cope.
That said, Ashley Banfield reported earlier on Saturday that her people had talked to the suspect's two aunts-by-marriage. One said she had to buy separate pots and pans because BK wouldn't eat out of anything in which meat had ever been cooked or served.
I would be very surprised if he could get that type of service at a Mad Greek diner.
(ETA and BBM: It occurs to me I have no way to link to AB's report. I watch YouTube on my Apple TV device: no links or addresses are given. I'll report this post myself to the mods.)