If you choose the .25/06, it will suffice as a moose gun, in a pinch, with the proviso that you choose a good bullet like a TSX or a Partition, so there's your versatility. If you shoot coyotes and intend to sell the pelts, soft points will prove a bit harsh, but Barnes makes a nice needle sharp solid that works well for pelt hunting. If you don't sell the pelts, any expanding bullet that shoots well in your rifle will get the job done.
Much that applies to the .25/06 also can also be said for the .243. There might be a better selection of 6mm bullets to choose from, but provided you can find a bullet that is appropriate to your needs, that doesn't really matter. A .243 might benefit from a faster rate of twist than a quarter bore, but again, this is dependent on your choice of bullets, and isn't absolutely necessary in order to enjoy the versatility that this cartridge has to offer.
The .223 is the least versatile of the lot. If its legal to hunt big game with a .22 centerfire in your province, the .223, loaded with a good bullet is suitable for deer, but not anything heavier. More so than with either the quarter bore or the 6mm, the .223 benefits from a fast twist barrel, so look for a 1:8 or even a 1:7. This will allow you to shoot the long TSXs when you go after big game. If you were looking for a dedicated coyote rifle, the .223 trumps either of the larger rounds, particularly if selling hides is a consideration. From an economic point of view, you will shoot more rounds of .223 per dollar than you would .243 or .25/06.