You can't get a legal visa to work in Thailand or China straight out of University any more.
Plus you have to remember that TEFL unless it is done at a higher level (universities, state 6th form equivalents) it is absolutely useless in terms of work experience.
TEFL in China is often you got a white face? You're hired, you are often hired as an entertainer and NOT as a teacher.
The experience is considered to be worthless.
Illegal is risky as illegal teachers now have bounties on them for dobbing them in (5000RMB which is 3 months wages for people so they do it). $10000 fine for school and you AND a life time ban from ever working in China for life and not being allowed to even visit China for 5 years.