The former Liverpool and Wales legend quit the Deva Stadium on Thursday, leaving the Cheshire club searching for their third manager of the season.
He finally called it a day after his assistant Mark Aizlewood was sacked against his wishes, just weeks after chairman Stephen Vaughan had publically declared his intention to sack Rush.
Rush said: "Various events have gradually made it impossible for me to carry on.
"I was already considering my position after the much publicised comments from the club a few weeks ago.
"Loyalty is vital in football and I am a loyal person. When a valued member of my team is dismissed against my wishes, I have no option but to follow my principles.
"People never know what goes on behind the scenes and that will remain the case."