Interestingly enough, it’s actually the square wave that is harder to perfectly recreate using the Fourier series. An infinite summation of sine waves is needed to avoid ringing.
The implications of this: the anti-aliasing (lowpass) filters used when capturing PCM audio will cause some distortion of a square wave, regardless of frequency. In practice, filtering and sampling at a sufficiently high frequency (even 44.1kHz) moves the aliasing well outside of normal human hearing (beyond 22.05kHz).
A 44.1kHz sample rate is sufficient to *perfectly* capture and recreate any signal with periodic components under 22.05kHz, so you need not worry about losing fidelity at 11kHz. The reason for FLAC vs. lossy encoders such as MP3 or AAC is due to the way the lossy encoders work. In essence, they use principals of psychoacoustics to remove information most people are unlikely to miss. In practice though, good ears can still (barely) discriminate 256kbps AAC from lossless, hence why formats such as FLAC remain popular with audiophiles.