As a USA citizen, if you enter the country and establish residency with the visa status 'Employee on Assignment' (Salarié en Mission) what happens if you and the company that brought you into the country decide to part ways (such as you leave/quit, not fired). Are you then forced to depart France?
A general principle in France is that you are never forced to leave while your carte de séjour or long-stay visa is valid. There is none of the automatic loss of status or super-short grace period you see in other countries, the authorities must first take action before you have to leave.
In practice, the préfecture does have the power to cancel your visa prematurely, in which case you would be notified by mail (and probably receive an obligation de quitter le territoire français at the same time, at which point you have 30 days to leave before facing real trouble) but this would take some time and I don't think they do it as a matter of course when people lose their job.
Mainly, the choke point is when your current residence title expires and you need to get or renew a carte de séjour (or possibly when renewing an autorisation de travail). That's when many people end up receiving an OQTF and/or staying illegally (sans-papier). For a permit as salarié and provided you lost your job involuntarily, you can actually get at least one additional year to look for an alternate job, possibly more if you qualify for longer unemployment benefits. Unfortunately, what little information I could find online suggests that this is not the case for a salarié en mission.
Worse, depending on your exact situation (and specifically whether your employer is a French business or not), you might not even be eligible to register as "looking for a job" (demandeur d'emploi), which means no unemployment benefits and, presumably, no right to work for another employer without applying for a new permission from scratch. So you would not be forced to leave immediately but finding something to stay after the expiration date of your current visa or transitioning to another status might be somewhat difficult.