Boyce, 26, has had two previous loan spells with the Mariners, making a total of 21 appearances for the club.
He was released by Scunthorpe in May, having failed to make an appearance for the League One side last season.
Browne, 20, has signed for an undisclosed fee after scoring six goals in 37 National League games last term.
The Antigua and Barbuda international joined the Shots in July 2015 following his release by Charlton.
Meanwhile, winger Nathan Arnold, 28, and defender Richard Tait, 26, have left Grimsby after rejecting contract extensions.
Find all the latest football transfers on our dedicated page.
Newly-promoted League Two side Grimsby have signed former Scunthorpe United defender Andrew Boyce and Aldershot winger Rhys Browne on two-year deals.