Calculating the 3dB frequency of an nth order low pass filter and nth order high pass filter
In general there is no straightforward analytical solution. As you know, you need to solve
$$\left|H(e^{j\omega_c})\right|=\frac{1}{\sqrt{2}}\tag{1}$$
for $\omega_c$, where it is assumed that the maximum filter gain equals $1$.
For Butterworth filters, the specified cut-off frequency always equals the $3\textrm{ dB}$ frequency. This is not the case for other filter types.
In general, you have to either estimate $\omega_{c}$ from the magnitude plot, or you solve $(1)$ numerically.

EDIT: For Chebyshev filters it's relatively straightforward to compute an analytical expression for the $3\textrm{ dB}$ cut-off frequency.
As an example I'll use a Chebyshev II highpass filter, as mentioned in the question. The magnitude of the normalized frequency response of the analog prototype filter is given by
$$\left|H(\Omega)\right|=\frac{1}{\sqrt{1+\frac{1}{\epsilon^2T_n^2(\Omega)}}}\tag{2}$$
where $T_n(\Omega)$ is the $n^{th}$-order Chebyshev polynomial, and $\epsilon$ is a parameter determining the ripple size in the stop band. The normalized $3\textrm{ dB}$ cut-off frequency $\Omega_c$ can be determined from the equation $\left|H(\Omega_c)\right|=1/\sqrt{2}$, which is equivalent to
$$T_n(\Omega_c)=\frac{1}{\epsilon}\tag{3}$$
From the trigonometric definition of the Chebyshev polynomials
$$T_n(\Omega)=\cosh\big[n\; \textrm{arcosh}(\Omega)\big],\qquad\Omega&gt;1\tag{4}$$
we can solve $(3)$ for $\Omega_c$:
$$\Omega_c=\cosh\left[\frac{1}{n}\; \textrm{arcosh}\left(\frac{1}{\epsilon}\right)\right]\tag{5}$$
Now we just need to transform $\Omega_c$ to the discrete-time domain according to the warping introduced by the bilinear transform:
$$\omega_{c}=2\arctan\left[\Omega_c\tan\left(\frac{\omega_s}{2}\right)\right]\tag{6}$$
where $\omega_s$ is the specified stop band edge of the discrete-time Chebyshev II high pass filter.
Note that we can also rewrite $(6)$ to determine the required value of $\omega_s$ given a specified $3\textrm{ dB}$ cut-off frequency $\omega_c$:
$$\omega_s=2\arctan\left[\frac{1}{\Omega_c}\tan\left(\frac{\omega_c}{2}\right)\right]\tag{7}$$
The following Matlab/Octave script illustrates the procedure of computing the $3\textrm{ dB}$ cut-off frequency of a Chebyshev II high pass filter:

% example filter specs of Chebyshev II high pass filter
n = 6;          % filter order
Rs = 60;        % stop band ripple in dB
ws = .5 * pi;   % stop band edge of Chebyshev II filter

% Cheby II: stopband ripple = ep / sqrt( 1 + ep^2 )
ep = 1 ./ sqrt( 10^( .1 * Rs ) - 1 );

% compute normalized 3dB cut-off frequency of analog prototype
Wc = cosh( acosh( 1 / ep ) / n );

% transform 3dB cut-off frequency to discrete-time domain
wc = 2 * atan( tan( .5 * ws ) * Wc );       % bilinear transform

I designed a discrete-time $6^{th}$-order Chebyshev II high pass filter according to above specifications. The resulting $3\textrm{ dB}$ cut-off frequency obtained from above script is $\omega_{c}=0.69\pi$. This is the correct value as shown in the plot below: