The essence of this solution has been found independently by a number of people. This particular machine spec is the one I came up with. It is followed by a walkthrough.
Mnemonics: I = "input" O = "output" S = "state bit" f = "full speed" t = "third speed" b = "bouncer" l = "left" r = "right" g = "gun" ' = [next state] Inputs: Ifl, Ifr, Itl, Itr ; unconnected looks like 0 Ibl, Ibr ; unconnected looks like 1 Outputs: Ofl, Ofr, Ohl, Ohr, Obl, Obr, Og State bits: Sfl, Sfr, Stl0, Stl1, Stl2, Str0, Str1, Str2, Sg ; initially 0 Sb ; Initially 0 except for the soldiers on the ends, where Sb ; is initially 1. If this is illegal, we could add a fourth ; set of I/O lines for the sole purpose of detecting whether ; a soldier has neighbors on both sides. Combinational logic: Ofl = Sfl Ofr = Sfr Otl = Stl2 Otr = Str2 Obl = Sb Obr = Sb Og = Sg x = Ifl && (Stl0  Stl2)  Ifr && (Str0  Str2)  Sfl && Itl  Sfr && Itr ; triggers the soldier to become a bouncer State transitions: Sfl' = Sb ? Ifl : Ifr  x Sfr' = Sb ? Ifr : Ifl  x Stl0' = Itr Str0' = Itl Stl1' = !Sb && (Stl0  x) Str1' = !Sb && (Str0  x) Stl2' = Stl1 Str2' = Str1 Sb' = Sb  x Sg' = Sb && Ibl && Ibr Starting action: Set Sfr and Str1 of the far left soldier to 1.
The initial state is stable, with all outputs 0. After the starting action, a 1 travels to the right through the fullspeed lines, and another 1 travels to the right through the thirdspeed lines. When the fullspeed 1 hits the far right soldier, it bounces and heads back to the left, meeting the thirdspeed 1 at the midpoint. If the number of soldiers is odd, they meet inside a single soldier, causing it to become a bouncer. If the number of soldiers is even, the two 1s cross between the two middle soldiers, which both detect that they are sending and receiving 1s on the same side, and both become bouncers. Whenever a soldier becomes a bouncer, it sends 1s on its fullspeed and thirdspeed lines on both sides. By symmetry, the two halves of the row will proceed in lockstep, as mirror images of each other. The row continues to subdivide until every soldier is a bouncer, then they all fire their guns on the next clock tick.
You can watch it happen with this wish script: toysoldier version 3.0.0.
The number of ticks from start to finish is 3*(n2)+1, or a little less depending on how the odds and evens work out. I haven't proved this, but you can get 3n analytically using a continuous approximation of the discrete system. The top level of the problem sends the fullspeed signal 1.5 times the length of the chain, then does a subproblem half the size. 1.5 * (1 + 1/2 + 1/4 + ...) = 3.
As noticed by Dan Garcia, it's possible to modify this solution to get arbitrarily close to 2n. Instead of cutting the problem in half each time, cut it into p parts, using signals at speeds i/(2pi) for i from 1 to p1, plus one fullspeed signal. The running time will be about (2+1/(p1))n. This adds a lot of complexity to the state machine.
