McGovern could not break into the Celtic first team after graduating through the club's youth system but the 23-year-old will have the chance to establish himself as the Tannadice number one.
The Northern Ireland squad goalkeeper will join on July 1 and has signed a one-year contract.
Shala, 19, has clinched a three-year deal with Craig Levein's side after the German impressed on trial and will also be available from the start of next month.
McGovern will rival Lukasz Zaluska for the goalkeeping role at United, while Andis may be joined by another forward before the season starts.
Levein said: "It's good to get another striker and I've got Noel Hunt, Jon Daly, Kevin Smith and now Andis in the squad. There's a chance I'll add another before the summer is out."
McGovern had little chance of breaking into the Celtic first team.
Poland international Artur Boruc has been firmly established as Celtic's first choice in goal for the past three years, with former Inverness number one Mark Brown serving as his deputy for the past 17 months.
"I want competition for Lukasz and Michael will provide that," Levein told the club's official web site.
"He is very well thought of at Celtic, but at a club that size first-team opportunities for young players can be difficult to get.
"In the last two or three years he has had Artur Boruc to compete against and as we have seen at the Euros Boruc is a top-class goalkeeper."