My solution for similar problem is for each item to keep a flag that keeps item state (i.e. checked or unchecked). In your case code will look something like this:
<ion-item *ngFor="let player of players">
<ion-label>{{player.name}}</ion-label>
<ion-checkbox color="dark" [(ngModel)]="player.checked"></ion-checkbox>
</ion-item>
This way you know what players has been selected and do what ever you need with this information.