The front passenger tire for my MDX is damaged beyond repair. Is it required that I replace all 4 tires together?

It depends on who you talk to. Certainly the manufacturer and tire companies will say so but their biggest worry is liability. A lot of people do not do this and don't have any trouble, not all of us have the deep pockets that are required to keep them happy.
For sure the tread depth or wear has to be the same on an axle so you should at least change them in pairs. Depending on how much tread is left on the driver's tire you maybe should consider replacing the pair, if it's nearly worn out. A used tire that will match up in tread design could replace a half worn tire. Tread design HAS to match on the drive wheels and should match on any axle.
Minimum tread depth, in Canada is 3/32 of an inch and is probably similar elsewhere.All tire dealers have a guage to measure this and should check the tire in 2 or 3 spots.
If your tire shop is giving you a hard time or refuses to work with you, find another shop. A "mom and pop" shop will probably be more user friendly than a large chain and the prices won't be much different. Just remember that a used tire is used and will not give you the same mileage as a new one. Also what do you think the tire shop is going to do with the 3 tires they are taking off? They don't throw them out if there is any tread left on them, they sell them. Hope this helps.