McKinney was a member of the Ireland Under-20 squad for the past two seasons, and played at the recent IRB Junior World Championship in June. The 20-year-old was awarded a development contract with Ulster ahead of this season.
Andrew spent the past season at the Castres Academy, as he completed a compulsory year in France for his French and Spanish Joint Honours degree at Queen's University.
The 21-year-old is a current member of the Phoenix Ulster Academy and plays his club rugby for Ballymena in the Ulster Bank League.
Darren Cave, who showed impressive form on his return from injury at the end of last season, is partnered in the centre by Ian Whitten.
At 26, scrum half Paul Marshall is the most senior member of Ulster's starting back-line, which also includes Conor Gaston and Adam D'Arcy in the back-three alongside Andrew.
The pack has a more experienced look about it with Tom Court, Andi Kyriacou and Declan Fitzpatrick making up the front row.
Tim Barker and Neil McComb have been given a chance to impress as a second row partnership. Chris Henry will captain the side from openside flanker and will combine once again in the back row with Pedrie Wannenburg and Robbie Diack.
New recruits Callum Black, Lewis Stevenson and Mike McComish are among 11 replacements hoping to see some action.
Academy players Adam Macklin (prop) and John Burns (hooker) are also in line to taste their first action in a senior Ulster shirt.
Dan Tuohy, Ian Humphreys, Nevin Spence, Luke Marshall and Nigel Brady are all being rested to give them extra time to prepare for the new season.
Meanwhile, Ulster remain without a number of other senior players who are with international sides in the build-up to the Rugby World Cup in New Zealand.
ULSTER (v Bayonne): Adam D'Arcy; Conor Gaston, Darren Cave, Ian Whitten, Ricky Andrew; James McKinney, Paul Marshall; Tom Court, Andi Kyriacou, Declan Fitzpatrick, Tim Barker, Neil McComb, Pedrie Wannenburg, Chris Henry (capt), Robbie Diack.
Replacements: Callum Black, Paddy McAllister, Adam Macklin, Jerry Cronin, John Burns, Lewis Stevenson, Mike McComish, Willie Faloon, Ian Porter, Paddy Jackson, Craig Gilroy.