It is necessary to visit the office of the respective Health insurance company and DEREGISTER from the system of public insurance in Slovakia. If not, you will be still considered as a health insurance holder (even if the employer deregisters you) and as such the company would claim paying the contributions. You should do so, within 8 days after you finish the employment in Slovakia and you should also return the Insurance Card (EU Insurance Card).
Documents that might be required: Work contract with the end date stated or residence card for the stay in Slovakia with final date stated.
More information: https://www.vszp.sk/poistenci/chcem-poistit/zanik-poistneho-vztahu.html (in Slovak only).
You are also obliged to notify the Foreign police of his leaving the territory of Slovakia and return the Slovak Residence card in person at the respective Foreign police office.
After the taxation period (that is one calendar year) the annual clearing of taxes is made by filing the tax return (declaration of taxes).
The deadline for submitting the tax return is 31 March every year (with the possibility of postponing this deadline for 3–6 months upon sending the notification to the competent tax office until 31 March).
Deregister the employee from the Social Insurance company within 8 days after the termination of employment (the respective university department shall easily deal with it).
The employer should notify the respective Labour office about termination of employment of an EU national within 7 working days thereof on the form: http://www.upsvar.sk/buxus/docs//SSZ/OISS/CUDZINCI/Priloha_11.rtf. Two originals of the form are required.