I assume it's not a workplace with a collective agreement (otherwise this should be handled by the union by some sort of informal or formal grievance procedure).
The employee can't sign away their rights and entitlements under the employment standards (usually) unless the law expressly provides for it (e.g. overtime agreements or concessions in collective bargaining). The employee is considered the (significantly) weaker party in the relationship and one public policy objective for the labour standards is to protect the employee.
Resignations must be voluntary, clear and unequivocal in most/all Canadian jurisdictions, otherwise it is treated as a dismissal. A threat to terminate with or without cause would make resignation involuntary. So it does not really matter if you "agree" to resign.
A settlement and release (agreement not to sue), providing more benefits than what labour law requires, might be valid or invalid depending on the jurisdiction. A settlement just giving what you are entitled to under the law or even less is not valid as there is no valid consideration.
Some jurisdictions like Quebec may explicitly disallow the employee from relinquishing certain rights
2092 The employee may not renounce his right to obtain an indemnity for any injury he suffers where insufficient notice of termination is given or where the manner of resiliation is abusive.
Civil Code of Quebec
In other jurisdictions, a release could be assessed based on its content and wording, if it is substantially unfair, if there was other undue influences that pressured or forced the employee into signing, if the employee was represented by a lawyer etc. (See Chow v. Mobil Oil Canada, 1999 ABQB 1026, which deals with the validity of release with respect to human rights claims, but broadly the principles also apply to other areas.)
So, if your jurisdiction would require more than a week of severance pay for termination without cause, the proposed solution would not be a valid agreement. It could potentially be valid if they still fire the employee with cause but provided additional compensation so the employee promises not to use (but whether one week's pay is enough to make the agreement fair would be for the court of decide.)