What does the icon with four squares mean

As far as I understand it, your Field Upgrade score is measured by your squads effectiveness with working together - attacking an objective works towards this, as well as reviving team-mates, etc. If you die your score drops beneath the level needed for the current Field Upgrade you'll see that message. As for if you can see it, for your actual score, I don't know. For the progress to the next Field Upgrade, it's the bar highlighted below (sorry for small size):

BF4 uses a "smart lean" system. The beta is a bit buggy with it, but the premise is that when that pops up, you are by a corner and holding the ADS button will cause you to lean around that corner, exposing less of yourself to the enemy. The best way to see it is to stand at the corner of a wall and play with it. It takes some getting used to but once you do, it will feel smooth.