We tow a Ford Fusion Hybrid, with has the same engine, drivetrain as the C max hybrid.
No, miles will not log on the odometer, nothing to worry about there.
One thing you do need to do ASAP, to save yourself a lot of headaches and difficulties later on: Install a charge line from your MH to the C max, to keep the battery charged up. Pretty easy to do, and will prevent you from running down the car's battery.
I actually put an ammeter on the battery to test it out. Found that even with EVERYTHING turned off, ignition in ACC position and tranny in neutral as it has to be when towing...Electrical system was still drawing around 3 amps from the battery all the time. If you have a supplemental braking system, in addition to any power it needs, the active braking system of your hybrid will also kick on every now and then, using up even more battery power.
We found after just 4 hours of towing, battery was weak enough it would not operate the power locks. Only took ONE trip of that happening, before I installed a charge line. :)
You NEED a charge line with these Ford hybrids, period. Good thing is, once you install it, you'll can pretty much forget about it, and never worry about battery charge again.