Gibbs phenonemon in real life
I figured it out! I created a MATLAB script which sums the first 1000 terms of the output's Fourier series with a 1Hz square wave input. Here's the answer:

Low-pass filters can cause the Gibbs phenomenon to occur, but these filters are not physically realizable. (This makes sense, otherwise non-causal behavior would be possible). Consider a brick wall filter. This is equivalent to truncating the Fourier series of the input to a certain maximum frequency. The results of the MATLAB script verify this:



As mkeith said in the comments, a perfect first order RC filter does not produce ringing. Instead, the filter just 'rounds off' the square wave:



The filter that I created was a first order filter, but parasitic elements, bad probes, etc. must have produced the ringing/overshoot, since as the result from MATLAB above shows - first order low-pass filters do not produce ringing.

However, second order (RLC) low-pass filter can produce ringing! The MATLAB script verifies this:



You'll notice that there is no non-causal behavior occurring here. This second-order filter is physically realizable (neglecting parasitic elements, but even then this behavior should be able to be approximately reproduced).

This means that the reason I was confused was simply because I didn't do the maths! I assumed that the Gibbs phenomenon would apply to a first order filter despite the fact that phase-shifting and attenuation occur in a first order filter which do not occur when truncating the Fourier series.