I have an E3 visa and on a W2 form. Am I eligible to switch to a 1099 form and still work for the same company?

For answer to your question read this
E-3 visa - Wikipedia, the free encyclopedia

YOu should also learn the difference between W2 wages and 1099 income:
1. W2 -- you are an employee. From your income social security and medicare taxes are withheld at 7.65%. You normally can't claim expenses associated with the job unless you itemize your deductions. Also your business related (job related) expenses are subject to 2% AGI limit.

2. 1099 -- you are independent contractor. On your income, you will pay social security and medicare taxes at 15.3%. If you have expenses associated with this job, you file Schedule C on which you record your income and expenses.

3. With 1099, the employer does not have to pay their part of employment taxes of 7.65% and some other taxes. They save headache of keeping another person on their payroll.