As the other posters said, first it is often illegal. Then if it isn't illegal to use FMJ's, they may be dangerous to use. Ever hear one whine across a frozen field? An expanding bullet would break up and disintegrate on hard frozen ground causing less danger to others. The FMJ can richochet and go a long way.
Additionally, they just don't offer quick enough kills. Most often coyotes will run after receiving a FMJ bullet. Coyotes are tough little critters and just don't lay down and give up after taking a bullet. Unless it is a bullet that scrambles the innards and causes severe internal damage and transfers enough energy to anchor them. Coyotes are soft footed and don't leave much in the way of tracks to help recover them. Especially in forest clutter like leaves, on grassy fields, or on hard frozen ground. No tracks makes it nearly impossible to track them to aid in recovery. That thick fur you are concerned with soaks up blood, especially from little pin holes such as a .22 caliber FMJ leaves in the hide. So you often won't have a blood trail. All that adds up to a lost animal.
Then there is the other side of the FMJ, the tumbling issue. Often the FMJ will yaw or tumble in flesh causing more damage than a hollowpoint/plastic tip/soft point bullet. Or, hit bone, such as the shoulder at an angle and you'll tear a gaping hole in the hide.
You are much better to choose a conventional hunting bullet and there is plenty of info in the archives here about .223 bullets to help you make an informed decision. Does that help explain it?