Magnitude and phase of frequency response from Z-transform
Its frequency response can be decomposed into a real part and an imaginary part as
$$
H(e^{j\omega}) = 1-e^{-j\omega} = 1-(\cos\omega - j\sin\omega) = (1-\cos\omega) + j\sin\omega
$$
The magnitude response is the modulus of the complex frequency response
$$
|H(e^{j\omega})| = \sqrt{\mathcal{Re}\{H(e^{j\omega})\}^2 + \mathcal{Im}\{H(e^{j\omega})\}^2} = \sqrt{(1-\cos\omega)^2 + \sin^2\omega} = \sqrt{2 - 2\cos\omega}
$$
where $\mathcal{Re}$ and $\mathcal{Im}$ represent real part and imaginary part respectively.
The phase response is the phase angle of the complex frequency response
$$
\angle H(e^{j\omega}) = \arctan \frac{\mathcal{Im}\{H(e^{j\omega})\}}{\mathcal{Re}\{H(e^{j\omega})\}} = \arctan \frac{\sin\omega}{1-\cos\omega}
$$