Updating this thread...
Accounting just received this error again. Apparently this issue still causes problems about once a month. We cut paychecks weekly (Monday mornings), so that's about a 1 in 4 chance of this error still causing problems.
To get actual work done, the person usually goes into the System Tray, right-click on Method Integration Engine, chooses Shut Down. This seems to fix it every time.
Today, however, the engine was not running when the error occurred...but QB still thought it was. Had to cancel out of the payroll check printing function altogether, causing the person to manually print each paycheck instead. PAIN. Luckily, *only* 17 employees...but still, that sucks.
I'm just telling the person to shut down Method Int. Engine BEFORE doing payroll to cut down the chance of these pieces of software tripping up on one another.