Highpass and lowpass filter phase-shifted by 90°
what should I consider if I wanted to ensure, that their exits are
  phase-shifted by 90°?


This is a problem facing relatively wideband I and Q circuits when trying to phase shift a modulated carrier in order to properly use IQ demodulation techniques.

The simple way is to feed your input to a circuit like this: -







simulate this circuit &ndash; Schematic created using CircuitLab

And choose R = R1 = R2 = \$\sqrt{L/C}\$. This ensures that OUT1 and OUT2 have a phase differential of 90 degrees across all frequencies.

So, with L = 250 nH and C = 100 pF, R = 50 ohms.

You need to scale L and C to suit the part of the spectrum you wish to use. With L and C as shown, it is optimized for a centre frequency of about 32 MHz (Fc = \$\dfrac{1}{2\pi\sqrt{LC}}\$)


  It should not be frequency dependent


The phase relationship between OUT1 and OUT2 is always 90 degrees but amplitudes do change with frequency as with any high-pass or low-pass fliter. Why not use a sim tool and check it out.