RE: Squares.. there's a reason why (most) all wargames used hex based systems, and it's not just for equilateral movement - it also allows for more precise facing calculations. When you're dealing with arcs of fire, hit locations where armor factors may differ based on exposed side, etc, it is really a requirement.
There are cases where hexes flat out suck - consider the facing movement costs in Battletech. Unless you are moving up & down or diagonal, moving horizontally costs a LOT of movement points as you have to make a facing change every time.
The only other real option for strategy games is to use NO tiles whatsoever, to make it completely open. When you do away with hexes, it *does* solve a lot of constraint problems for line of sight, and other issues. For line of sight you just draw a line and if anything intersects, you have no line of sight.. However you have a great deal of other, new concerns (such as unit stacking, object collision, and so on.) As soon as you do away with a structured system for location & facing, you have to be a lot more tolerant of loose rules and arbitration thereof.
My original plan was to build a game that used a conventional smooth terrain map. However, I quickly realized that if I did that, it added a whole SLEW of physics questions I wasn't prepared to answer (no rules to back me up), such as calculating a slope, how much a unit slows or speeds up when traveling up or down a slope, what the maximum slope a unit can move on would be, and so on.
In the end I decided to start a hex engine because everything (well, MOST everything - there's some arbitrary rules dealing with line of sight when a line goes perfectly through two adjacent hexes). Anyway, it can be defined in one way or another with an algorithm without bringing physics in to play, which means I can ultimately complete the engine more quickly and get to the good part (playing!).
New obligatory screenshot follows. :)
Need to go back and do some abstraction work and general cleanup before I do more features.. trying to figure out what to do with the horizon at the moment, then on to clouds, I think.