During the civil rights movement BYU became the target of many boycotts and protests by players of other football teams due to the LDS's prohibition against African Americans becoming church leaders (since changed). The most famous incident was the "Black 14" protest which devastated the Wyoming football team for years.
http://www.sltrib.com/byucougars/ci_13728556
San Jose State had actually started the boycotts against BYU. Tony Jackson led a group of African American SJSU players who refused to play in the 1968 BYU game. The players thought they would lose their scholarships over the incident, so it was a very serious decision for 18-21 year old students from mostly poor backgrounds to take. Jackson, by the way, went on to become a Marine Corps general. He was the SJSU commencement speaker a few years back.
Memories of these events kept diverse, inclusive schools like Stanford away from BYU for a long time. I have a fuzzy recollection that Bill Walsh--always very conscious of issues related to race in sports--nixed talk of a BYU game.Â
The LDS church has changed a number of its policies, and BYU's past civil rights incidents have been largely forgotten. The work of sociologist Darron Smith (who is black and Mormon), however, has reawakened the spectre of racism at BYU. I can't vouch for him one way or the other other than to note that BYU apparently fired him. For what it is worth, here is an editorial discussing his analysis of race at BYU and the case of recruit Jabari Parker:
http://arewomenhuman.me/2011/04/28/race-...revisited/
and here is his site:
http://www.darronsmith.com/
Whether you buy into Smith's current critique of BYU, there is the obvious conflict between Stanford's inclusive policies on sexual orientation and those of the LDS church, which operates BYU. Before now gay college football players have generally not disclosed, but by 2020 I would guess that they will be mostly out. This could set up similar conflicts and incidents as occurred with African American players in 1968-70.
I think a BYU game would be a mistake.