In any circumstance, employers should urge sick staff members with influenza-like symptoms to return home immediately and contact a health professional at that time. This should involve phoning the person's normal doctor or nurse or a specially designated centre to seek further advice, rather than the patient calling in without prior notification.

При любых обстоятельствах работодатели должны рекомендовать больным сотрудникам с гриппоподобными симптомами немедленно вернуться домой и в это же время связаться со специалистом в области здравоохранения. Для этого, чтобы избежать визита пациента к врачу без предварительного уведомления, необходимо позвонить врачу сотрудника, медсестре или в специально назначенный медицинский центр, чтобы получить дополнительную консультацию.

" This should involve"   =/=  "Для этого"

можно объединить с пред. предложением -

.....связаться со специалистами/сотрудниками учреждения здравоохранения, а именно: позвонить своему лечащему врачу/мед. сестре или в  специально назначенный медицинский центр,  чтобы получить дополнительную консультацию , вместо того, чтобы приходить лично без   предварительного уведомления 

и сразу же обратиться к... 

"This should involve" зд. вполне себе = "Для этого"; иначе можно сказать "Обращение должно осуществляться посредством телефонного обращения к...". Канцелярит, конечно, но текст вроде бы официальный.

employers should:

a) urge sick staff members with influenza-like symptoms to return home immediately; and

b) [employers should also] contact a health professional at that time.

Similarly, from the way it's written,

'This should involve phoning the person's normal doctor or nurse ......' means

'This should involve [the employers] phoning the person's normal doctor or nurse ......'

Logically you would expect it to say that the staff member should contact his/her own doctor etc rather than the employers contacting the doctor -- but that's not how it's been written. The English could have been written more clearly to avoid any ambiguity.

If it's a legal document you may need to clarify who exactly is supposed to do the contacting -- but if it's a non-legal document, User's 'Обращение должно осуществляться посредством......' may be better, as it avoids specifying who exactly should do the contacting.

That's why, if it's a non-legal translation, I would use User's suggestion or a variant of it -- to protect yourself against accusations that you've mistranslated who exactly should do the contacting!

....employers should urge sick staff members with influenza-like symptoms to return home immediately and  contact a health professional ..... -  sick staff members should return .... and contact....

This ( contact b y   sick staff members)    should involve phoning the person's ( sick staff member's)  normal doctor or nurse ...

Source:  http://www.gtha.com/sites/default/files/Pandemic.pdf

"...In any circumstance, employers should urge sick staff members with influenza-like symptoms to return home immediately and contact a health professional in the manner advised by the Public Health Agency of Canada on its website at that time. This should involve phoning the person's normal doctor or nurse or a specially designated centre to seek further advice, rather than the patient calling in without prior notification. "

employers should  urge sick staff members (what to do?)  

a) to return home immediately; and 

b) [ urge sick staff members to] contact a health professional. 

Isn't it possible to take it this way?

If it's intended to refer to the staff member, it's written in a slightly clumsy (unnecessarily wordy) way.

I accept that, logically, you'd normally expect the staff member to contact his/her own doctor -- but if the author meant to say that, they shouldn't have written:

'This should involve [???who???] phoning the person's normal doctor or nurse or a specially designated centre to seek further advice, rather than the patient calling in without prior notification.

but instead:

'This should involve (them* / him/her / the staff member / the patient) phoning their [not 'the person's'] normal doctor or nurse or a specially designated centre to seek further advice, rather than calling in ......'

* where 'them' means 'him or her', ie the staff member.

It may be that the author intended to write what you've both written -- but if so, he/she has written it in a slightly clumsy way. That's why I think it's ambiguous.

Если немного расширить контекст, то пазл складывается: "and contact a health professional in the manner advised by the Public Health Agency of Canada on its website at that time. This should involve..."