Would the following frequency response be considered linear phase? I know that linear phase is when the phase response is a linear function of frequency. However, the phase response is only linear from ~0.3 so I'm not sure if it is considered "linear".
The phase response is clearly non-linear, as mentioned in a comment. Note that phase jumps at the zeros of the frequency response occur depending on how the phase is defined, but these phase jumps don't make the phase non-linear. However, in the given example the phase is clearly non-linear at low frequencies.
You could also look at the corresponding impulse response. If the impulse response has a finite length and if the coefficients are symmetrical or anti-symmetrical with respect to the center, then the phase is linear.
Another method is to look at the pole-zero diagram of the transfer function. If the system is causal then all poles must lie at the origin and all zeros must either lie on the unit circle, or they must occur in quadruples $z_0$, $z_0^*$, $1/z_0$, $1/z_0^*$.