Bone conduction and in-ear microphones pick up the Bone-conducted (BC) speech, which has traveled through bones and soft tissues. This BC speech is less sensitive to surrounding noise than Air-conducted (AC) speech. Therefore, there is an interest in recording this signal. However, the intelligibility and quality of these microphones are known to be a limit to their use. Previous work has noted confusion between vowel sounds in terms of intelligibility. To evaluate the quality and intelligibility, studies rely on subjective and objective tests. This paper aims to determine if the standard objective methods for rating speech quality and intelligibility can be applied to BC speech recorded through these microphones. In order to estimate intelligibility, a subjective test based on vowel recognition was compared to STOI and to a new criterion based on second formant frequencies of oral vowels. For speech quality estimation, MUSHRA and PESQ tests were compared. Results show difficulties in using objective methods instead of subjective ones. Therefore, for ongoing studies, it is suggested to use subjective methods to evaluate speech quality and intelligibility.