I have a Norstar Meridian with M7310 phones. When calling out using the speakerphone, the microphone doesn't work. When trying to switch to hands-free during a call, it puts the line on hold.

Sounds like your telephones not programmed for handsfree.
Programming is done on 7310, 7324 and 7316 type sets.Enter by pressing the feature key followed by **266344.When prompted to enter the password enter 266344.
The screen should say TERMINALS&SETS. The 4 top line/feature keys are now used for programming.
At the terminals&sets prompt, press the show key (top right arrow). The display should now ask for an extension number. Enter extension number you want to change to handsfree.
Press the show key (top right arrow)
The display should say LINE ACCESS
Press the next key (bottom right arrow) .
The display should say CAPABILITIES.
Press the show key (top right arrow) .
The display say foward no answer.
press the next key 3 times (bottom right arrow)..
The display should say handsfree. Press the show key.(top right arrow) change from none to auto
Press the release key when finished