George was born in Salsburgh, North Lanarkshire. He started his career with Plean Juniors, before moving to Albion Rovers, in 1920, where he became a regular in the first team.
When John Bell, was transferred from Dundee to Albion Rovers, in July 1922, George made his way to Dens Park. It was reported that he was “a strong tackler, 5ft 9in, almost 12 stone.”
He made his debut in the opening match of the 1922-23 season, a home league fixture against Aberdeen, which ended in a 1-1 draw. He garnered favourable comments in the newspaper report, where it was said “There was not harder working half-back afield than Greenshields, if he has a fault it is carelessness in parting with the ball.”
In a hotly contested match away to Clyde on the 29th August, Dundee threw away a 2-0 lead losing the match 4-3. Greenshields was “sent to the pavilion” late in the game for comments made to the linesman. In September he had to appear in front of the Referee’s Committee of the S.F.A. with regard to his sending off for “insulting language to the linesman” he was severely censured.
It seems that George could not find the form to stake a regular place in the first team, following a ten match run at the start of the season. When he did get another opportunity, in mid-December, as on other occasions “he was of little account.”
In December 1922, after making eleven appearances, George was transferred to Motherwell, there he made thirty-six appearances. He finished his football career with Dumbarton at the end of the 1926-27 season, after seven appearances.
George’s family had a coach business in Salsburgh, Greenshields Bros. where he worked. He was injured in an incident at the garage in 1926. One of the buses went on fire, and his neck and were hands scorched, as he tried to stop the supply of petrol to the engine.
Away from football he was a racing pigeon enthusiast, becoming one of Scotland’s most celebrated pigeon breeders. There were few trophies that he didn’t win in the sport . He also gave lectures on the subject.
He died 17th February 1960.