I don't know what square icon your refering to unless your speaking about the download dmg. Normally the applications are placed in your HD application folder. Anything else on the desktop is left over from the dfownload process and can be placed in the trash.

To place a shortcut in the dock locate the application in the hard drive. (not your home folder) From the application folder you locate the firefox planet icon and drag it to the dock. What you get is an automatic icon on the dock and the application remains in the application folder. But just to make sure send me a picture of the part of the desktop in question.

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