This isn't surprising.
So, let's look at one of these peaks: What does having a peak there mean?
Right, the FFT is just an implementation of the DFT. A peak at frequency $k$ means that the sum
$$S[k] = \sum_{n=0}^N s[n]e^{-j2\pi \frac kN n}$$
is large.
It's pretty easy to imagine how that happens:
See thee $e^{jx}$ term in that sum. That's a complex sinusoid of relative frequency $\frac kN$. Now, what that sum does is simply taking a vector containing (equidistant samples of) that sinusoid, and forming the dot-product with the time-sample vector $s$.
Geometrically speaking, we're projecting your signal onto a sine of a given frequency, and get a complex value, whose amplitude tells us "similarity" (phase of that complex value is phase of the signal, but we don't care about that for now).
Let's reduce this to the real-valued case; it's easy to go back to complex later:
Now, what happens when you draw you rectangular wave, and on top of that, a cosine that has its maximum right at the center of each positive halfwave, and then do a point-wise product at equidistant sampling points?
Right, you get a very high value, because you're effectively calculating something like the integral of the abs() of the cosine.
Now, jitter the edges a bit. Does that change things much?
No. The edges fall in places where amplitude of the cosine isn't high to begin with. Sure, you lose some energy in your overall sum, but it's not going to make that high product value go away.
That's exactly what you observe here. Shifting the edges around a bit does reduce the energy in the harmonics of the square wave's frequency a bit, but not much. (You can see that you're actually distributing energy from the harmonics onto your wideband noise floor, for example, at the harmonic third harmonic from the left.)
I don't know why you're doing this, but for example in switch-mode power supplies, you need to switch a transistor on and of pretty often. Now, doing so with a square wave will introduce strong spectral peaks (as you've demonstrated!), which will lead to (potentially harmful and illegal) RF emissions.
So, some switch-mode power supplies use what is called a spread-spectrum clock, which doesn't only jitter a few edges – it inverts (seemingly) random half waves! Only on average that clock is half-time on, half-time off, but in between, it's a binary pseudorandom sequence.