Does a square (or any non-sinusoidal) wave a definite wavelength?

If you get from a Fourier transform the spectra of the square wave you get many wavelengths (and of course frequencies), and it has a bandwidth. Freq and wavelength is defined wrt to the basis waves, which are sinusoids. See below about other basis waves.

If you use it differently it'll be easy to have people confused, or just say you are wrong. The spectrum of the square wave is the delta function convolved with a sync function. In radio and EM transmission and reception they try to filter out the sidelobes (and any freqs outside what you want), and usually the square wave is rounded by some smoothing function.

In QM it's even more critical because the sinusoids are the eigenfunctions of the Hamiltonian (or some physical and important operator that you use with eigenfunctions as the basis for expanding any arbitrary solution. The eigenvalues or for the Hamiltonian the energy, are then we'll defined. A square wave or a limited wave train will not have a definite energy.

This is also reflected in the uncertainty principle in QM, and it is equivalently true for any wave. If your wave last infinitely long your uncertainty in the arrival time is infinite. Then the energy uncertainty has to be zero. In classical waves you need a long train to measure it's frequency accurately, equivalently to QM.

Now, you can choose other basis waves, or in QM, eigenfunctions of any operator (with some constraints) and expand on the basis of a those. The eigenvalues, or values you could actually measure and get a number, will just not be energy. Similarly for classical waves, you can choose mathematically any basis you want. In some signal processing work people use so called wavelets, and other things.