Former St Mirren centre-half McGregor put pen to paper on a one-year deal with the SPFL League One champions on Wednesday.
The 28-year-old has suffered two cruciate ligament injuries during his four years as a player in Paisley.
And some Gers supporters have expressed scepticism the defender will be able to hold down a place in Ally McCoist's side.
But McGregor, who was playing Junior football with tiny Arniston Rangers just six years ago, is no stranger to defying the odds.
He bounced back after being released by Cowdenbeath as a youngster and has returned to full fitness after two serious injuries.
He said: "I want to come in and prove my worth to the fans and to the gaffer and also to the players as well. The hard work starts here.
"I've been given the opportunity, the same as I was given at Cowdenbeath and the same as I was given at St Mirren.
"Hopefully I can come in over the coming months and prove to the fans that I'm worthy of wearing that jersey every week. I hope to do that."
Rangers captain Lee McCulloch formed a solid partnership with Bilel Mohsni in the heart of the Rangers rearguard last season.
And McGregor, who played nearly 40 games for the Buddies last term, has admitted it will be difficult to displace the duo.
But the Edinburgh-born player is confident he can feature for McCoist's men in the second tier and help them return to the top flight.
He said: "Both Lee and Bilel are great players. McCulloch has played at the highest possible level. He's obviously getting on a bit, but with his age he has experience which is great.
"I just want to go in and prove my worth. If the gaffer thinks I can play then I can play.
"I'll certainly learn and certainly become a better player. There are Scottish internationals here and guys who have played at the top level. I'm looking to learn, more than anything."
Doubts continue to surround the financial future of Rangers and this week it was reported that the Ibrox club would try to raise £8million with another share issue.
But McGregor, who was out of contract after leaving St Mirren, stressed he had no qualms about joining the Glasgow giants.
He said: "I have never delved into the financial side of it. I have switched off from the ins and outs. I am here to play football.
"With the support that I have seen Rangers get in the Third and Second Division I don't think there is a chance they would ever go under.
"They are greatly supported and the fans wouldn't allow it to happen. It didn't cross my mind, to be honest."