Their non-compete is in all likelihood not enforceable because the constraint of what is essentially minimally skilled laborers that do not have access to things such as patents, formulas etc. is merely stifling competition, rather than being for any business related reasons. But the business can baressentially anyone from their building.
For a non-compete agreement to be enforced an employer must show reasonableness, in that (1) it has a legitimate business interest sufficient to justify enforcement of the non-compete clause; (2) that the clause does not cause the former employee undue hardship; and (3) that enforcement of the clause will not be harmful to the public.
Under this three part test, courts consider several factors when determining whether a non-compete agreement is reasonable, including: (i) how long the restriction lasts and the geographic area that the restriction covers(generally no more than 2 years and reasonable local area will be reasonable),(ii) whether the employee was the sole contact with customers, (iii) whether the employee possesses confidential information or trade secrets, (iv)whether the covenant operates to bar the employee's sole means of support, (v)whether the covenant seeks to stifle the inherent skill and experience of the employee, (vi) the likelihood that the employee can find other employment if the restriction is enforced and (vii) whether the benefit to the employer is disproportional to the detriment of the employee.
The underlined factors support the agreement not be enforced, but that they were the sole caregivers is one factor that would work against them. In any event, any private entity can bar any former worker from entering the premises if it wishes...regardless of enforceability of non-compete. Thus, if you want them to continue you to care for them in those facilities, there is no chance of that. In another facility however, it is probably possible, but they would have to have the non-compete ruled unenforceable by a court.