Somewhat like littlejay x360a mentioned:
Squad Rush is the way to go, but rather than going into a full lobby, try and go into a complete empty server (adjust the filter so that you will find some). Simply join one of the maps which aren't that popular (but it really doesn't matter where you are joining, just making it easier) with at least one friend. Now all you have to do is to cross your fingers for no enemy joining the server during the whole match.
After joining the server for the first time with 1-2 friends, it will tell you that it is going to start in a few seconds. After this happened, the person who wants to have the MVP ribbon just simply attack the 2 MCOMs to gain the most points and end the round quickly.
For MVP#2 a friend of yours is going to attack the MCOM, and for MVP#3 both friends are attacking one MCOM each, so that you will be on the third place.
This is by far the easiest and fastest way to get the achievements. All you have to do is to be lucky for no one joining, but that shouldn't be the problem. I was doing this with my friend a few days ago and we managed to go for the MVP achievements within 15 minutes...
Hope this helps for those which can't get it on full servers :)