What is the eligibility criteria to work abroad after completing post-graduation in engineering?

It depends actually on the rules of that particular country. As some countries do offer PSW (Post study work visa) options like Ireland, Canada, USA, France, Australia, etc while other restrict like UK, Germany, Sweden, etc. So, you might have to come back as soon as you finishes your post-graduatio

There is no criteria for working in abroad except graduation as you are a fresher. You should have some experience in the field and after that, you can go abroad for a good salary package in the field of your interest.