I am very sure about this.
The timers show time until:
(box) = Time until a building on a base has a package to be collected
(box with excalmation) = Time until a building on a base has full packages and can not get any more.
If you hover over either of these timers, it will show which base will be affected.
These icons will display in the base "status" bar below the timer area if there is a package waiting to be collect or if the maximum amount of packages has been reached and no more will be created for a building. (The "battery" icon means resource-storage is full... this also stops package production)
The OP stated:
"One of my box icons on one of my bases has a yellow triangle exclamation icon over the box. The other base does not."
This means that one of his bases has a building that already has its maximum limit of packages waiting to be collected.
To collect packages inside base-view, all you need to do is click on them.
